Can using a trainer damage my bicycle? I know that using my road bike with a fluid trainer will wear out the back tire more quickly than regular road riding, but does it have the potential to actually damage my bike (frame, gearing)? While not a super expensive machine, I do like it quite a bit and it is my only bike. 

While on the trainer, I notice (1) every little noise from the drivetrain and (2) some flexing of the (aluminum) frame as I pedal. It's quite possible that these things happen on the road, but it's too loud to hear them.

Should I think about getting a trainer-only bike?
It is possible but quite uncommon. The trainer holds the bike upright while you throw your weight around, and the frame is not really designed for that. Bike fames are very strong with regard to vertical forces but nowhere near as strong for sideways forces. Twisting the bike between the rear axle and the handlebars is kind of ok (because it's designed to take a twist between the handlebars and bottom bracket) but it's something to minimise. You're more likely to see a fatigue failure than a simple overload.

If you are moving your upper body as you pedal you should try to learn not to do that. Wind trainers are all about sitting still and pedalling, not about practicing sprints. You should avoid getting out of the saddle, and try to minimise upper body movement. Or you may need to go with rollers rather than a trainer. 

You definitely shouldn't bolt the trainer down - having it free to move almost guarantees you won't break your frame. The break I've seen was from one bolted to a concrete floor and the rider was out of the saddle when it broke.

As Billy said, your rear skewer will suffer cosmetic damage. Trainers often come with a cheap steel skewer for this reason.