I just got this game off of GOG after having played it years ago, since I read that they fixed the crashing problems. That's awesome that I can play it now, but I still suck at this game, haha.
My question is, is it advisable to play this game using keyboard only, and if so, how do I keep from getting killed every time? Whenever I'm in combat with two or more aggressors, I always end up getting beaten to a pulp. I don't have a joystick, and the missions I keep failing are Gatekeeper and the one where you fix the FTL Relay. Any suggestions?

Gatekeeper is a tough mission. Took me 3 tries to complete the mission, and I worked on the game!
You can play with keyboard only, but obviously joystick is easier. Use the A and Z buttons to control your speed rather than the throttle. Soften enemies up from a distance using ripple fire missiles before you go in close and attack.
If you're in trouble try to LDS away. You might get inhibited but often you can get enough distance to make some emergency repairs. Sometimes prioritising an LDS repair is the only way to survive.
For Gatekeeper specifically try to stay behind the L-point. You can then blast the enemies as they jump in their most vulnerable area - the engines, where there's a gap in their shields. Also any enemies will have turn around and approach you to attack, bringing them in range of the activated gunstars.
Try to face your enemies, your shields are strongest at the front.