Would they qualify as stars? Would they be able to maintain integrity with the relativistic change of mass (across its volume) while still running the fusion process?
A very speedy object however doesn't look like this: objects it passes have vectors which take them away from it. So while it can warp spacetime with that mass, it can't form an event horizon - there are vectors which lead things away from it and thus can escape.
[EDIT: I had to go do some reading to check up on this and it actually has an analogue to electromagnetism: moving objects develop gravitational effects perpendicular to their direction of motion, essentially this is one of the origins of gravitational waves - but it's also important to the overall point, which is that even if you had infinite gravitational acceleration perpendicular to your direction of travel, you've got numerous other degrees of freedom where this doesn't happen and thus escape vectors - so no event horizon though you might suffer a lot of damage and be spagettified anyway]
Black holes only work when you have a point force exerting uniform acceleration in all directions towards itself - beyond a key point, no attainable velocity ever results in an object not moving towards you.
Whereas a speedy object has an obvious one: just jump off the back of it (i.e. accelerate slightly against it's vector) and your trajectory now points you away from it.
For the star, from its point of view its mass has not changed at all. So yes fusion still works.