You already seem to have your accepted answer, but your question edits indicate some uncertainty. Let me see if I can offer an easier way to see it.
First off, what is the question? As I read it, your question is essentially,
Suppose we take special relativity and add, in one privileged reference frame, $N$ one-way portals which can teleport matter/information to points on corresponding worldlines, so that the 4-displacement from the portal entrance to exit is either locally simultaneous or future-pointing. Can we use these portals to create a grandfather paradox?
And the answer is no. (It can be trivially "yes" if they are two-way portals that are not all simultaneous; it remains "no" if they are two-way portals that are all simultaneous.)
Light bubbles and the two light cones of an event
To understand why, we simply look at the light cones.
An event, say a supernova, emits a lot of light. To anything which hasn't seen this light, that event has no causal power yet. We describe this as a "future" light cone which is an expanding "bubble" of photons in spacetime that announces to the universe that the event has happened. Since it expands at constant speed, if you project-out one of your dimensions of space and use that dimension instead to visualize time, these photons describe a cone as the circle expands uniformly in time: hence the name, "light cone". Now every point in spacetime "within" that cone is in the event's objective future: those points in space have observed the event, so there is a definite time-order. There is no definite distinction between where these events are spatially because a spaceship, bound to stay slower than $c$, could have had just the right sort of trajectory to happen to be at the locations of both events when they occurred; its reference frame locates both coordinates "right here", hence they may have happened at the same or different places.
The event also has a "past" light cone that you can get by extending all of those light rays backward through the event to the time before the event. This has an intuitive meaning too: the points inside this "past-pointing" light cone are objectively in the supernova's past: the star, pre-supernova, has observed all of these events by the time that the supernova has occurred. Then the space between these light cones is a sort of "relativistic present" for the event: these points in spacetime are objectively not at the same place as the event, but they may be at the same time or not depending on your frame of reference. The "local presents" for the supernova are hyperplanes in spacetime which pass through the supernova and live between these two light cones; given any such plane, there exists some reference frame which thinks that these spacetime points in the hyperplane are simultaneous; the hyperplane is a "present", but localized only to that reference frame.
If you finally understand these definitions and you want to kill your grandfather or otherwise send a signal to yourself before you send it, the perspective in terms of light cones becomes very simple: To get a grandfather paradox, you have to get information from an event into the past-pointing light cone of an event.
Why the answer is "no".
Relativity says that every reference frame is correct for analyzing the happenings in spacetime and that they all agree on the general topology of these light cones etc.
In particular the privileged frame above is always a correct way to view the whole spacetime. In this privileged frame there is a monotonic-invariant: no matter how information passes through these portals, the local time coordinate in this reference frame starts positive and keeps increasing. But the past-pointing light cone in this reference frame only has negative time-coordinates. So it's really simple: you cannot get into the past light cone using these portals. Some reference frames see this communication as the future having affected the past, but it is not a robust enough effect to actually generate a paradox.
With that said, other reference frames will still have paradoxes to deal with. The simplest one is temporary violations of conservation of energy: you throw a ball through the portal and some reference frame sees a time where two balls coexist briefly.
But it matters that it's all simultaneous in the same reference frame.
Now let's do the reverse. Suppose you have just two portals; they transmit information faster-than-light in two different reference frames. Then some of these configurations allow a grandfather paradox. The exact calculation of when this occurs is a little tedious; let me phrase it like this: in the simplest case of co-inertial portals, each portal consists of two worldlines which can be written $r_{0,1}^\alpha(s) = c_{0,1}^\alpha + s~t^\alpha $ connecting points at the same $s$; hence $r_1^\alpha(s) - r_0^\alpha(s) = c_1^\alpha - c_0^\alpha$ is always a spacelike vector of constant size $c$.
With no loss of generality we can do a bunch of stuff with this. Choose some reference frame and determine which FTL portal is "more" future-pointing in that frame, then boost into the coordinates where its $c_0$ is at the origin and its $c_1$ is simultaneous, at some point $q^\alpha = (0, c, 0, 0).$ This means we only need to mess with one $r_{0,1}^\alpha$ line and its corresponding $s$.
Now in this reference frame, the time-coordinate of $r_1^\alpha(s) - r_0^\alpha(s)$ is negative; then we solve $|r_0^\alpha(s) - q^\alpha| = 0$ for $s$ to find the "closest" portal $r_0(s)$ we can enter from $q.$ We then get to violate causality if the corresponding vector $r_1^\alpha(s)$ is timelike past-pointing; the portal must emerge into the past light cone of the origin. That's the essential criterion for having a causal loop with two fixed portals.
If you can just teleport into a simultaneous moment elsewhere in your own "present" coordinates, then (perhaps with some restrictions on your acceleration and teleport-distance) you can cause grandfather paradoxes willy-nilly: you teleport, accelerate towards where you came from, "tilting" your local-present hyperplane underneath the point in spacetime you teleported from, then you can teleport into your own relativistic past.
In any curved spacetime we can still talk about local reference frames that are small enough scale we can ignore the curvature. We also can ask if there are closed timelike curves (CTC) which basically is asking whether we can time-travel to our past selves. CTCs are strongly thought to be impossible in reality.
The universe is thought to be spatially flat, but the spacetime as a whole is curved. CTC's are impossible: at each point in spacetime you have an "age of the universe". To be precise, this is maximum path-length (proper time) a geodesic could have between the big-bang singularity and said point. Any time-like or light-like curve is moving in the direction of increasing age of the universe; this is just as strong a concept of "future" and "past" as in flat spacetime.
With a single warp-drive you don't have CTC's. But you can still get CTC's with multiple warp-drives. Suppose you build a warp-drive on Earth and send it out into space. You start with an (almost) flat initial-condition and then generate a strongly curved spacetime (your warp bubble). Starting from a flat spacetime (or for very large scales from the spacetime of the universe), is much more physically realistic than starting from any other spacetime. You have to make your weird and wonderful curvature from an "empty canvas" !
With a warp-bubble, the highly curved spacetime is on a small scale. This allows us to glue two bubble spacetimes together so long as the ships don't get very close to each-other. If we consider two Earths, moving relative to each-other, that each make a warp-drive, we can set up the system to generate CTCs. This is one reason we suspect this to be impossible.
There is another reason to suspect making warp drives is impossible: Geodesics would have to diverge in some region, which is an anti-gravity effect. Neither matter nor light can make anti-gravity (antimatter has positive mass just like matter). The "attractive gravity only" rule is more precisely defined as an energy condition and at least one of these is violated by warp drives. Violating certain energy conditions would make the speed of sound faster than light which also allows for time-travel paradoxes.
In general, no known solution with CTC's is physically realistic. They either involve infinitely large systems that cannot be setup from an "empty canvas" or violations of energy conditions. For example, the Kerr metric concentrates it's energy condition violation in it's singularity. Real black holes are thought to lack this feature and be much deadlier instead.
Best Answer
The thing about the speed of light $c$ is that it's not just a number associated with a certain type of particle. While we could talk about the mass of the proton, and there would be no problem assuming non-protons had greater or lesser masses, the value $c$ is an entirely different beast.
$c$ is an intrinsic property of spacetime itself, not of the particles in spacetime. You wouldn't expect there to exist anything (particle, signal, information) that you could insert into spacetime with the property "changes the nature of spacetime for itself."
The above is meant to undermine the "if it's just a property of everything we've found so far, we haven't ruled out finding things without this property" line of thought.
For a more concrete demonstration of what goes wrong when you alter the structure of spacetime as we know it, take a look at the "tachyonic antitelephone". This demonstrates how any abstract communication faster than light leads to causality violations as observed by even not-faster-than-light observers.
There are many variations on the thought experiment; here's one of them (with the math worked out in the linked article): $A$ is moving away from $B$ with speed $v < c$ in $B$'s reference frame. $A$ sends a faster-than-light message to $B$, who responds in kind with a reply. The problem is for sufficiently fast (but not faster than $c$!) speeds $v$, $A$ will receive the reply before sending the message. Faster-than-light anything begets time travel.
Since it gets repeated far, far too often, I'll also counter quantum entanglement arguments here. Quantum entanglement does nothing in the hypothetical "why can't this communicate faster than light?" scenarios other than guarantee that the two particles will "collapse" to the same (or opposite, or orthogonal, or whatever) states when either is observed. The only quantum mechanical aspect of the whole thing is the fact that the eventual collapsed state can't even exist before the measurement (it's not a hidden variable).
But the correlation -- the thing you want to rely on to communicate faster than light -- could be achieved entirely classically. Take a red marble and a blue marble, put them in a bag, and draw one out randomly without looking. Lock your marble in a box, and hand the bag to someone else, who also doesn't peak. Send the person far away. Then, look at your marble. If it's blue, you instantly know the other person has red. But you transmitted nothing. All quantum mechanics does is make it so who has which marble isn't pre-determined.
This shouldn't come as a surprise, because the only notions of space and time in quantum mechanics are the ones from spacetime itself. Quantum mechanics doesn't come equipped with some independent notion of distances. So whatever structure spacetime has applies to quantum things just as much as marbles and people and signals.
Everything said here stays true even in general relativity, by the way. Enabling things to move faster than $c$ still leads to causality violations. Even wrapping things in a "warp bubble" and moving that faster than $c$ leads to causality violations.