Personally I will suggest that you don't spend over 150$ on a HU. You can get what you need without spending that much. Something like this would do you right:
Kenwood KDC-X497 (kdcx497) Single-Din Car Stereo w/ Built-In HD
Now I'm not an expert on head units because I haven't bought one in years so I don't know what's out there really... I just did a search on
Sonic Electronix... So I'm sure someone else could get you one with a little better features... But if you decide to get that one then you probably won't be disappointed unless you already know a lot about car audio.
One of the most important things to look for in a HU is the preamp voltage. 4 volts or higher is recommended. The one I linked has 3 sets of 4V outputs which means you can connect an amp for your sub(s), an amp for your front speakers, and if you want you can amp your back speakers. But most people don't even have back speakers anymore and neither do I so putting an amp on your backs is up to you.
And about that radio you found in your closet, yes that could work. But it only has one pair of preouts so you can only run one amp on it. Which might even be fine for you if you don't plan on amping your front doors, but I'd suggest you do.
-Now to decide on your subwoofer situation. You said your space is pretty big so I'd say to keep in budget we have a couple options. Without knowing your actual space I'm gonna say do a single 18... An Obsidian Audio 18 costs 190$ plus shipping and will take 600 watts RMS daily. If you can not fit a box big enough for that then I'd say get 2 12s from Obsidian. Almost half the box size is needed to run them properly and they each will take 600RMS daily so 1200w total. Anything else will cost a considerable amount more money, like 100$ more. You can get an amp that'll do 1200 watts for like 150$.
So that's 110$ for the HU, 300$ for the subs, and like 150$ for the amp. You'll then need to get wire and either build the box yourself (which idk if you can) or have someone build it for you. That can be like 150 - 200$. So lets estimate that at 760$.
A little over budget but if you think you'd be happy with just 1 12 then you can save about 100$ and get a Sundown Audio SA-12 for 190$. And then you'd only need a 600 watt amp. So you'd save about 150$ maybe. But remember, that's only if you can't fit an 18. If you can fit an Obsidian Audio 18 then I'd def do that rather than an SA-12. It'd be a lot louder.