Welcome. Good pickup too.
Is a tune worth it on the stock turbo?
Yes, but do it straight after you install the other breathing modifications.
I'm planning on getting an Xforce 3" catback exhaust, but is there any significant power benefit to going with a turbo back system instead?
The power benefit comes from the forward section (dump pipe to cat). The iconic/better sounds come from the cat-back section. If your budget does not permit a tune, just get the cat-back section. If you want a power increase, go the full exhaust (or even a larger front section adapted to stock cat-back) AND get a tune.
...sounding a bit better (boxer rumble and some turbo woosh noise)
For intake noises some people go for a so-called "cold air intake". Design and quality vary. A lot of them turn out to be a hot-air-sucking pod filter in the engine bay. With the stock turbo you will not be making any more power, but the intake sounds will be louder. Almost certainly requires a tune after installing. Probably the worst "bang for buck" modification in my opinion [*
braces for incoming disagreement from others*].
A much cheaper way to get more intake noise is removing the "3 litre milk bottle" white plastic resonator from the bottom of the stock airbox. If you stop there it will suck a bit of hot engine bay air but sound better. If you extend that bottom section (pictured below) with a 76mm silicone hose (and/or aluminium and/or bends) so that air is being drawn from a cooler area near the foglight you might draw some cooler air.

Will be keeping the car for a year at most, so have set a budget of $2k max for mods.
That budget will be tight, so either the budget or the wish list will need to be flexible. If cash is king, just throw on a cat-back, delete the milk bottle resonator, spend any change on replacing worn items (and/or tightening up suspension) and be done with it.
Good luck with it all.