Can somebody recommend a soldering iron and/or tips for jobs like this one here?
Any iron with temperature regulator would work, as for the tips I'm getting good results with
these (I couldn't find on Amazon US but at least you can get the idea).
it just takes a long time to heat up.
It has to, especially when you desolder. Let the iron reach the proper temperature, then clean the tip on a wet sponge until it shines. I usually turn the iron on at the max temperature and the lower it according to my needs. Desoldering usually requires higher temperatures, in my experience.
How do I tell what the previous switch is in the matrix?
You follow the net from the switch pad to another. In your case, looks like the spacebar switch goes directly to the microcontroller, which makes things harder.
You might want to try your luck and connect the spacebar to a random switch to see what happens. Put the switch back in its place, connect your keyboard and then bridge its pins to another switch's, then activate the switch and see what happens. It's a terrible approach to a solution but it might work.
I did some reading and found what I thought was the previous switch in the matrix- the L Alt. I loosely wired them and when I pressed the space all I did was close the circuit for the alt switch. I found different techniques for wiring so I may not have even done it correctly. I'd be much oblidged if you could tell me the switch and technique I should wire this space bar to.
Read the previous bit, just make sure you bridge the switches with the correct polarity. Meaning left pin to left pin, right pin to right pin.