I've read a bunch of articles/guides to see if I could find answers; while I've learned a lot, I could use a little more help.
Long story short, the rear wheel on my ****ty road bike (which I acquired a little over a year ago for $100) is shot. My friend had a random wheel lying around that's the right size, but the bolt portion isn't long enough and I can't figure out how to remove the gears (or whether it's a freewheel or a cassette - it also doesn't spin freely).
This is his wheel:



And this is my broken one (3 or 4 spokes are broken):



I've established that my gear set is a freewheel, but I'm not sure which exact tool I'd need to remove it. All I know is that it says "maillard" on it, and there doesn't seem to be a standard maillard removal tool.
It seems to me that my options are as follows:
1. If it's possible, transplant my freewheel to the other wheel (and maybe the bolt along with it?) and buy only the removal tool. I'm pretty sure now that this one isn't possible, but it can't hurt to ask.
2. Buy a new wheel and transplant my freewheel to it, buying the wheel and the removal tool.
3. Buy a new wheel and a new freewheel.

Which of these options is the best and which is even feasible? I'd like to spend $40 or so total if possible, but will obviously have to spend more if I go with option 3.
What's the best place online to buy this stuff? I've noticed some is available on amazon, but I figure there may be somewhere better/cheaper.
My goal now is to fix my bike (as crappy as it may be) for as cheap as I can so that I both don't have to buy a new one and learn some more about bike repair in the process.

Thanks!

Can somebody please at least tell me why I may or may not be an idiot for having posted this?

Does the sprockets on his wheel spin at all? It's hard to make out from the pic, but it looks like he might have some funky two speed fixie arrangement on there.
If it's a "real" fixie I don't think you'll be able to fit a regular freewheel on there, but if it's something cobbled together out of a originally freewheel hub it should be OK. Lockring tool or hammer + drift punch. Regular hub it should unscrew counterclockwise.

What you are referring to as "the bolt" is probably the axle. Axle and cones needs to match up for the bearings to work. Not so good odds of sucessful random matches.

I first thought that you might have the dreaded helicomatic hub, but probably not.
There are at least 3 different splined freewheel pullers. Either swing by a bike store and have them remove that one for you for a few bucks, or get us a solid measurement of the diameter of the splined cavity. Vernier caliper is recommended.

You may not find an actual Mallard freewheel tool but there's only so many patterns and one of those patterns will work. Or just take the wheel into a bike shop and ask them which it is. Or go to www.parktool.com and look at the pictures of their tools to select the one you need.

Your wheel has a solid axle and his has a quick release axle. The difference is that your axle uses the nuts to hold it on and his uses a quick release skewer to clamp it into place. The skewer is missing on his wheel. But again bike shops could help you with this aspect.

You say that on his wheel the gearing doesn't move freely. Can you spin it for multiple turns and it's just gritty or does it only turn a couple of times and then locks? It's hard to tell from the picture because the hub is so fuzzy but it's very odd that there's only two sprockets on that hub. It may well be that it's some funky two speed fixie thing as mentioned. If so then the gears would turn a few times on the threads and then lock if the lockring is loose. If they keep turning then it's got a freewheel of some sort that you can swap out. But that means you need TWO tools unless you're lucky and the one freewheel tool fits both freewheels. You'd also need a skewer to get this wheel back into use since as mentioned it's not likely that the cones and spacers will fit between the two wheels.

There IS another thought. Remove your freewheel and replace the spokes that are broken along with any others that are badly loose and then have the whole wheel properly tensioned and trued. From your post it's likely that you should get this done by the local shop or find someone that has done this before so they have the right feel for it or bad things can easily happen. But since it sounds like there's nothing wrong with your wheel other than a poor tension and true doing this would make it fine again and likely better than it was before. And get the front treated the same way since very likely it's poorly tensioned as well.

Have the bike shop remove the old freewheel, drop it in the scrap bin,
get a new freewheel and the matching removal tool . straighten out the axle and spoke problem with new parts.
they will give you stuff ... I can just type words..

The gears on the extra wheel don't spin at all, and I didn't notice any place I'd be able to put a lock ring to remove the gears. Based on what you guys have said I suspect that that wheel is useless to me.
Do you really think its worth trying to replace the spokes on my wheel rather than just buy a new one? It looks like I could get one for $35 or so on amazon that I suspect would be better than what I have now.

In my experience the cheap wheels that I've gotten for some repairs or just because I'm being cheap all needed to have the spokes stress relieved and re-tensioned and trued again right out of the box. Buy a $35 new wheel and you'll very likely be right back where you are now within a few months at the most if you ride often. Spend that same $35 to have a shop that knows their stuff replace the broken spokes and any dodgey spokes with new ones and properly tension and true the wheel and it'll be reliable for years to come.

Often bike shops build wheels during their slow times. If you can get a budget wheel that was built by them it will likely be tensioned and trued to a much higher standard than the Amazon factory machine built wheel. It may cost you $15 bucks more but the money would be well worth the extra reliability. If you can find such a wheel or buy the new wheel from the shop and talk them into stress relieving going over the tensioning of a factory machine built wheel from them for an extra $10 you'd be far, far ahead of just pulling one out of a box and thinking that you're done. Such machine wheels WILL "settle" and need re-trueing and re-tensioning as they stress relieve themselves in use. Ignore that fact and you'll be right back where you are now. Having the shop do it up front will avoid that. But no shop will do it for cheap if you bring in a wheel that you didn't buy there. But buy from them and they should do it for cheap or maybe no cost since it's a fairly quick thing to do on a new wheel.

Then it sounds like a fixe set up to offer two different ratios.

You wouldn't have to put a lock ring anywhere, that's what's probably there to keep things from humorously unscrewing themselves while riding.

If you look at your blurry top picture, left side of the hub. First there's the bigger sprocket, then there's the smaller sprocket. Then just outside that is what I just can convince myself that it looks like a ring with a few notches cut into its circumference. That'd be the lockring. It even appears to be a short part of hub base material protruding outside the (assumed) lockring, which gives you a fighting chance of determining thread direction.

If my interpretation of the blurry pic is right, then some careful introduction of Mr Hammer and Mr Drift Punch, or possibly Mr discarded Screwdriver to the notches will take the lockring off.
After that the sprockets should come off with a chain whip. There are ways of improvising around that if you haven't got one.