Hello I will help you with your question,
I think you may be on the right track with the struts causing the rough ride. You may even have a frozen strut as well, I have seen this happen. I would replace the mounts at the same time as you do the struts as it costs little more and the labor is the same to do them at that time.
Of course be sure the tires are inflated properly
You don't list the mileage on this car but go ahead and flush the transmission just be sure the shop uses genuine Hyundai SP3 trans fluid. Do not let them put a one "one size fits all" fluid in the transmission. I would have the axles inspected along with the mounts (there are four) your description is tough for me to pinpoint a source for the noise. I would have to hear it to be sure of what is making the noise.
The timing belt needs to be changed every 60,000 miles and the sparkplugs at the same interval, injector/intake cleaning will help with the drivability.
Let me know what other questions you may have about this