Personally I prefer going up against SAM sites in Falcon and taking airborne stuff out with AIM-120s rather than dogfighting. Dodging SAMs in Falcon by diving under them while inverted and forcing them to lose their lock with the terrain is really great fun! Nevertheless, I did practice dogfighting the Falcon F-16 a lot simply because I wanted to be able to fight well if I got into that situation.
The rudder most definitely CAN help you win dogfights in Falcon.
In dogfighting (especially in Falcon) it's vital to practice maintaining the best corner speed without afterburners (somewhere between 340 and 440 knots is about what to aim for in an F-16), therefore, well-coordinated turns are a must, which means a rudder control is a must. If you can hold that best corner velocity and maintain around 5g, you can pretty much out-turn anything in Falcon (eventually LOL). Do that and it will be easy to get your lift vector onto the bad guy, to enable you to take a shot. It does require very precise flying to not bleed off speed when doing this kind of malarkey(this is where the rudder is useful). Even so, you still find you need to drop the nose every so often in a low yo-yo maneuever to maintain the speed. But you might be able to get away with a bit of bottom rudder to get the nose below the horizon to assist in maintaining speed instead of doing it inverted with the lift vector, so again, the rudder would be useful here too.
The Falcon 4 manual (both the original and the Allied Force PDF) has a good tutorial and accompanying practice mission in the sim which will demonstrate all this stuff and teach you what to do. Even when you are doing it right it still takes a lot of turns to get on the tail of the Russkie jet you fight in that tutorial (or at least it does when on full realism settings), but if you can do it without using the afterburner too much, you'll kick a lot of people's asses when fighting online, as many people who fly flight sims online haven't got a clue about maintaining corner velocity without the 'burner on!
Incidentally, maintaining the best corner velocity without using full power was what many of the Harrier pilots in the Falklands did, and had done so before when they flew the F4 Phantom. This is one of the main reasons why they completely kicked the Argentinian pilots asses in that war, because the Agentinian pilots were not able to turn with them without relying on 'burners and using up fuel while at their operational range limit. Consequently, they'd turn for home when the bingo fuel warning came on and get a missile up their tailpipe while trying to break off the engagement.
I'm pretty sure Falcon will let you use automatically coordinated rudder (alfhough I'd never use it myself as it wouldn't let you make crossed controls maneuevers such as a sideslip), so I imagine it is possible to fly the Falcon F-16 without a rudder control, but I personally would not want to.
You most certainly can do close-in dogfighting in Falcon, it's actually one of the best jet sims there is for it. But it does require you to practice all that stuff above which I mentioned in order to do it (not really a big deal, as even practicing dogfighting is fun). This is one of the reasons why the Falcon 4 manual has such a legendary reputation as one of the best manuals ever though, because it has pages and pages of stuff on how to do all that kind of thing properly in it.
If I were you, I'd just get an intermediate joystick with a built in twist rudder function and throttle, then worry about getting something fancy when you've decided whether you really want to spend the cash on it. If you really are in the mood to spend more cash right now, you'll do a lot better with a less flashy stick and TrackIR, than a really flash one and less flexibility when it comes to looking around. Remember, you have to be able to see something first before you can dogfight it!
Chock