Do the fundamental forces change in an object at rest vs the same object moving at relativistic speeds? Say this object is a magnet, would it have the same magnetic strength if it were traveling at.9c, less, more?
Similarly, would gravitational attraction be the same?
Would the other fundamental forces change?
 A: All the fundamental forces would have the same strength. 
Gravity: 
It is a misconception that mass creates gravity, it is stress-energy. Now you are asking whether gravitational attraction would be the same. Even a photon, that has no rest mass, does have stress energy, and does have gravitational effects. Now this photon is traveling in space with speed c in vacuum when measured locally. It still does have the corresponding gravitational effect that can be calculated from its stress-energy. So yes, a photon, traveling at speed c in vacuum when measured locally, does have a gravitational field around it, and this gravitational field travels with the photon at speed c. 
Now let's see a particle with rest mass, like an electron. Traveling at speed 0.9c the electron would have its own gravitational field, that can be calculated from its stress-energy.
EM force:
You are asking whether a magnet or a charge traveling at speed 0.9c would have the same EM force as the charge let's say at rest. The answer is yes, if you measure the EM force the electron traveling at speed 0.9c has, you would get the same charge (which happens to be the elementary charge), as if it would be at rest. Now there are no experiments that check the EM charge of an actual electron at rest, it is calculated theoretically. But you would get the same EM charge.
Now you can try to measure the magnetic force or electric charge of a non-elementary charge (like a composite object), at rest, and then let's say try to speed it up to 0.9c. You would still measure the same EM force for the object.
Weak force:
there are no experiments currently, that would measure the strength of the weak force. I do not know of any such experiment that would measure it at 0.9c speeds at all.
Strong force:
It is the strong force that holds quarks together to form a neutron or proton, and it is the residual strong force, or nuclear force that holds the neutrons and protons together in a nucleus. Now if you take a composite object, like a proton or neutron, and try to measure the strong force, there is no experiment for that, but you can try to calculate it theoretically, and at rest you would get the same measurement as if the proton or neutron would travel at speed 0.9c.
There are exotic atoms, like the pionic atom, where the electrons are replaced by pions, and then the pions are held to the nucleus by the strong force. They use these to measure the strong force between the pions and the nucleus.
But anyway, the speed of the object will not change the strong force inside it.
Please see here:
https://tel.archives-ouvertes.fr/tel-01674426/document
