Question 1: Yes. You are right.
Q2: Hmm... You don't have to think of it that way. You are relying too much on visualization, which in my experience can lead to difficulties later on. To understand these constructions and concepts, it is best to have alternative ways of thinking about them.
For instance, you can say that a cone is just the cylinder, but to have a continuous function from cone to somewhere, values must agree along the top ring; i.e. $f(x,1)=f(y,1)$ for all $x, y$ in your space. Continuity of $f$ will force nearby values to also be close to one another.
It can be that in some situations you require that up to half the height your cylinder be untouched, and only after that allow the points to "suffer the deformation." Talking about a topological space only, you don't have to impose any smoothness criterion, and up to homotopy equivalence, or even homeomorphism, a cone can be embedded in many weird shapes (upper hemisphere can be viewed as the cone over the circle.)
Q3. Please specify what direction your $f$ is taking? To check its continuity check the criteria I mentioned above. Intuitively, you would push down on the tip by your palm to flatten out the cone into the disk, so send the tip, all of the collapsed ring, to the cented of the disk. Level zeor ring, the bottom, sent to the out most circle of the disk.
Quotient spaces are not always very easy to visualize. What's worse, there is no uniform method that applies to let you visualize them all. Like most things (in and out of mathematics), you simply have to experience a lot of quotient spaces to learn how to understand, and sometimes visualize, them.
For this quotient space $X^*$, first let's look at some important subspaces. First let's ignore the two points $(0,-1)$ and $(0,+1)$. Notice that $(\mathbb R - \{0\}) \times \{-1\}$ and $(\mathbb R - \{0\}) \times \{1\}$, which one can think of as "two copies of the line minus the origin", are being identified into one single "line minus the origin", a subspace of $X^*$ that I'll denote $L$.
Next, $X^* - L$ has just two other points, which I'll denote $0^- = \{(0,-1)\}$ and $0^+ = \{(0,+1)\}$. The union $L \cup \{0^-\}$ is homeomorphic to $\mathbb R$, and the union $L \cup \{0^+\}$ is also homeomorphic to $\mathbb R$.
So, one can think of $X^*$ as a line with two origins, which is what people actually call this beast. If you want to try to visualize it, imagine that you are simply looking at the actual real line, except that when you try to focus on the origin you have double vision. That's about the best one can say for this example.
Best Answer
Let $X$ be a topological space, and let $A\subseteq X$. $A$ defines the following equivalence relation $\overset{A}\sim$ on $X$: for $x,y\in X$, $x\overset{A}\sim y$ iff either $x=y$, or $\{x,y\}\subseteq A$. The quotient space $X/A$ is defined to be the same as the quotient space $X/\overset{A}\sim$.
Added: To put it a little differently, the partition of $X$ induced by $\overset{A}\sim$ is $$\{A\}\cup\big\{\{x\}:x\in X\setminus A\big\}\;.$$ The topology of of the quotient space is defined as follows: $V\in X/A$ is open iff $\pi^{-1}[V]$ is open in $X$, where $\pi:X\to X/A$ is the obvious quotient map.
As Alex Becker mentions in the comments, this notion of quotient is somewhat different from the notion used when some algebraic structure is present, as in the case of quotients of groups, rings, vector spaces, etc.