I'd really like to play this and stuff like Privateer but was wondering what a good stick would be. I don't want to break the bank or anything but I'd just like to be able to enjoy these games with fun controls.

Logitech 3D Pro if you find it at decent price (in theory it's $30 or less), or the basic Thrustmaster T-flight (not the HOTAS versions) is also good enough (TM) and should go for a similar price.
If you don't want to break the bank for these old games avoid the T-flight HOTAS or similar "faux hotas" joysticks with the full throttle lever incorporated instead of the small wheel or slider because:

1- most games have enough keyboard commands that the "HOTAS" (hands on throttle and stick, that is a flight control system that allows to have all relevant functions on throttle and stick) nature of these peripherals is moot because you'll be reaching for the keyboard all the time;
2- most old games before windows 95 don't support a throttle axis no matter what, the oversized lever would just get in the way when you need to reach the keyboard;
3- they are big but have few buttons for their overall bulk.

Privateer in particular doesn't lend itself to throttle controls even with mapping software trickery (I did something with CH Control Manager but it's... not great).

Personally I play these games with a CH Product HOTAS (568 Combatstick USB and Pro Throttle) but I am much more into them than most people.

I ordered a T16000M would that work?

Yes, of course.
I didn't mention it mostly because from what I know it's already on the costly side (also it has my personal pet peeve of a button layout that puts too many buttons on the base and too few on the stick itself) but it's a good choice.
With that one you can also use the Thrustmaster software to map keyboard commands to the extra buttons the older games don't support more than two or four, CH Products has a similar software and it's quite handy.

Does this game work on win 10? And if it does, which type of control stick do I need for playing?

It does, I had to install win10 on my machine since the last time I posted in this thread and it works fine.

Regarding flightsticks: see above.

No need for super fancy stuff especially considering that most games don't support throttle controls even though you can make the games work with that kind of thing somehow with some software trickery (Joystick Gremlin is a popular choice when proprietary software suits fall short).
These games were mostly conceived to be played with right hand on the stick and left hand on the keyboard, HOTAS systems are just some extra icing on the cake.

If you'd rather, I originally played through X-Wing/TIE Fighter with a ye olde Logitech Not-Super-Nintendo-But-Totally-Is-A-Super-Nintendo-Controller in the 90s. You can get away with something similar if you want to drop $50-60 for a comparable 8BitDo controller on Amazon. Sure, it's not a flight stick but you don't desperately need a flight stick for these games.

Not advisable, he might as well play with only the keyboard (IIRC the DOS games allow that).

I get using weird controls (I finished Half-Life and Jedi Knight using only the keyboard in my days) but at that point he might as well drop those 50/60 dollars on a T-16000m joystick (the version without pedals and throttle handle), there are even some that will do the job and are below that price range easily.