No, there would be no detectable - and surely no dangerous - changes to the Earth's orbit.
Just for the sake of an argument, imagine that we double our coal reserves by bringing coal from another place - and even some of the precious metals are too expensive to be brought by spaceships at this moment. ;-)
The Earth's coal reserves are something like 1 trillion tons which is $10^{15}$ kg. Let's bring this amount from another celestial body - it's about 10 orders of magnitude less than what we can do now but let's imagine we double our coal reserves in this way.
The Earth's mass is $6\times 10^{24}$ kg, so the coal reserves are approximately $10^{-10}$ of the Earth's mass. Now, if all the extraterrestrial coal landed by a slow speed relatively to the Earth, the Earth's velocity wouldn't change at all; only the mass would increase and the heavier Earth would continue along exactly the same trajectory as before.
But now, imagine that the coal lands at some speed, e.g. the safe speed that the space shuttles used to take. It's about $350$ km/h. If you don't get approximately this low, there's a risk that your coal will burn in the atmosphere.
So if $10^{-10}$ of the Earth's mass has a relative velocity that is $350$ km/h, and let's imagine that all the momentum will go in the same direction - we could make the coal space shuttles land at different places if we wanted - then the velocity of the Earth will change by $350\times 10^{-10}$ km/h which is $3.5\times 10^{-5}$ m/h or $10^{-8}$ m/s. The speed of Earth around the Sun is approximately $3\times 10^{4}$ m/s, so we only change it by a trillionth. Correspondingly, the eccentricity of the Earth's orbit could change at most by one trillionth. We would have a hard time to detect this change.
Obviously, you would need to increase the amount of resources you bring roughly by 10 orders of magnitude (which means by 20 orders of magnitude relatively to what we can achieve today) to produce any threat for the Earth. But even if you brought the whole Moon here to Earth, $7\times 10^{22}$ kg (eight orders of magnitude heavier than the Earth's coal reserves), there would be no significant change of the orbit because the speed of the Moon is approximately the same as the speed of the Earth - they're bound together. Well, the shape of the Earth could change a bit if we tried to incorporate the Moon too quickly. ;-)
You would have to bring a big fraction of Mars to the Earth (Mars is both heavier and has a substantially different speed) to change the eccentricity of the Earth's orbit by a significant amount and I assure you that this will remain in the realm of science fiction for many, many centuries if not forever. If you brought 1/3 of Mars to the Earth, you would also have to build mountains that are 1000 kilometers high - more than 100 times Mount Everest, around the whole Earth. And it would still not be too dangerous as far as the orbital characteristics go. Of course, there could be a danger for the people who suddenly have 1000 kilometers of rock above their heads. ;-)
Your question clearly seems to be an artifact of the unscientific doomsday scenarios that have been presented as science in recent years - e.g. the doomsday caused by CO2 in the atmosphere. Even relatively to the atmosphere, our additions of CO2 are negligible - we're changing the number of molecules in the atmosphere by 2 parts per million (0.0002%) per year. But the atmosphere is just one millionth of the Earth's mass, so our annual CO2 emissions only redistribute something like 2 parts per trillion of the Earth's mass every year. Clearly, all those changes are irrelevant from a "mechanical" viewpoint and they're arguably irrelevant from a climatic viewpoint, too.
For an object in low earth orbit (at 100+ miles above the earth's surface) the speed needed is about 17,000 miles per hour. Even if a trebuchet could achieve that speed on the earth's surface, you would have at least three problems:
The object would IMMEDIATELY burn up in our dense atmosphere. Think about the space shuttle which is going at orbital speed when it encounters the very tenuous atmosphere at very high altitudes. It needs special heat resistant ceramic tiles due to the heating caused by a very tenuous atmosphere. If the angle at which the first encounter the atmosphere were too steep it would completely incinerate. So there is no material that you could use to build the satellite that would prevent it from immediately burning up.
If you could magically make all the atmosphere disappear, you still could not launch a satellite with a trebuchet from the surface of the earth. Well you could, but it would only complete less than one orbit. If you got the right speed, it would start out on a nice elliptical orbit, but the ellipse would bring you back to the launch point coming up through the crust of the earth. In other words the ellipse will pass through the earth such that in less than one orbit you will impact the earth's surface again. To successfully launch, the satellite would need to have some kind of rocket motor onboard so that once it got to an appropriate altitude, it could change the velocity direction to be in an orbit that doesn't intersect the surface of the earth.
The last problem that will make this Trebuchet impossible is the mass and required strength of the arm that will connect the heavy weight to the pivot point to the satellite. I suspect that making this arm strong enough will make it too heavy to work. So, for now let's assume the arm has zero mass and infinite strength. Then if we assume the heavy weight falls in say, about 1 second at about 1G, then to get the satellite to 17,000 miles per hour, the acceleration of the satellite would have to be 25,000 ft/sec^2 which means it would accelerate at 780Gs (so humans would be killed for sure). That would mean that the length of the arm to the satellite would have to be 780 times longer than the short arm to the heavy weight. So if the short arm were 10 feet, the long arm would have to be 7,800 feet which is 1.5 miles. I think you can see that the arm requirements would make this totally impractical if not impossible. For this to even work, the heavy weight would have to be greater than mass of the satellite times the long arm length divided by the short arm length by a very large factor (to insure the heavy weight falls at about 1G). If we assume a 100kg satellite, then in this case that means the heavy weight would have to be something like 10 or 100 times (7800/10)*100 kg - thus something like 780,000kg to 7,800,000kg. Imagine the strength of the arm that is required. Then think about how heavy the arm would be and how that would make all of these requirements even more impossible since a heavy arm would greatly decrease the acceleration of the satellite.
So, no it CANNOT be done...
Best Answer
Energy Needed ($E$) = Potential Energy at L1 ($V_{L1}$) - Potential Energy at Earth's Surface ($V_e$)
$$V_e = -Gm(\frac{M_e}{r_e} +\frac{M_l}{LD - r_e}) $$
$$V_{L1} = -Gm(\frac{M_e}{d_{L1}} +\frac{M_l}{LD - d_{L1}}) $$
where $m$ is the transported mass, $M_e$ is Earth's mass, $M_l$ is the Moon's mass, $LD$ is the center to center Earth-Moon distance, $r_e$ is Earth's radius, and $d_{L1}$ is the distance from the center of the Earth to L1.
The formula above it for a two body system, along the center line of such a system. You could replace $d_{L1}$ with a smaller distance if you want to go less than all the way to L1 along this line.
In the formula for $V_e$ substitute $r_e + 100km$ for $r_e$
This subquestion doesn't have a specific answer. Even if a zero mass payload is assumed, the propellant and structure holding the propellant have mass. The chemical nature and mass of the propellant, mass of the inert sturcture, and arrangement of stages would be major factors. Multiple stages are advantageous to reduce the mass during flight by eliminating no longer needed inert sturcture.
If fuel is not used to make a soft landing on the Moon, this would definitely increase efficiency.
Addtional considerations:
None of the above considers the gravitational potential of the Sun. If you don't mind crashing into the Moon, launching when the Moon is between the Earth and the Sun would minimize the amount of energy needed. This would add a third term involving the Sun's mass and distances to the Sun to each of the potential energy equations.
As previously suggested by user "I like Serena", since Earth is rotating about its own axis, a rocket launched from Earth will have an initial velocity component due to this rotation. It is optimal to launch from near the equator to take advantage of the maximum velocity due to rotation, as well as greater Earth diameter/less gravity. Launch should be timed such that the rotational velocity component is directed to the Moon as much as possible, at which time the direction of the Moon will be generally eastward.
The L1 point is calculated considering gravitational potential and centrifugal force of a body in the rotating frame. The point of maximum gravitational potential along a line joining the Earth and Moon would be a somewhat different point. It would be more correct to find the maximum gravitation potential along this line and use that potential energy value, although it should be similar to the potential energy to get to L1.
For more information on low energy transport to the moon, without using Hohmann transfer, and without crash landing, see Low Energy Transfer to the Moon. An alternative low-energy approach was used by the 1991 Japanese Hiten mission.