With the correct rocket setup would it be possible to thrust around in a circle, like you would do donuts in a car, to generate centrifugal gravity?
Thrust won't generate gravity, but it will produce acceleration which may be indistinguishable from gravity to an occupant. Yes, it's possible to simulate gravity by having a spaceship constantly thrusting to travel in a circle, but it would be an awful waste of fuel.
Rotating a large object, or pair of objects connected by a tether, to simulate gravity is more efficient because, once started, it will continue to rotate with no further input of energy.
For a space based object, there is absolutely no problem with the energy need - you can get that from solar panels, RTG (fission heat to electricity via thermoelectric effect). Each comes with a different set of tradeoffs, but they all work now in one spaceship or another (Mars rovers, satellites, deep space probes, ...).
What you can't have in space is reaction mass (as in Newton's action and reaction). Even the most mass-efficient thruster (electrically accelerated ion thrusters) use up their fuel - and in deep space you can't get any more (except for the - currently theoretical - Bussard Ramjet).
Also, any practical gravity (let's say half Earth gravity) is high in terms of fuel burn rate. I didn't find fuel burn values, but the trans-lunar injection was a 350 seconds burn for 3 km/s delta-v, or about one gravity. The fuel flow was (according to this page: https://www.nasa.gov/centers/marshall/pdf/499245main_J2_Engine_fs.pdf) about 530 lb/sec, or 250 kg per second. In monetary terms, that's more than half a million dolar per second - but the real issue might be one of availability not one of cost (rockets can only lift so much).
So, while the theoretical answer to your question is yes, the practical one right now is no.
Sure, though I can't think of a reason you'd want to do that. Burning the engines in any direction produces acceleration that feels like gravity. Given the tremendous cost of bringing more fuel, if you're in a spacecraft you are probably trying to get somewhere.
Spinning in a circle gets you nothing. There isn't a tether or similar to supply the centripetal force, so you'll be supplying it with the engines. And after you've spun around in a circle your net change in velocity is zero.
More likely, you'd just burn prograde or retrograde, in a straight-ish line, turning only to keep the ship aligned with your orbit. Besides using fuel to get you to where you are going rather than burning in circles, the occupants will be more comfortable as they won't need to deal with significant Coriolis forces.