Honestly, with your budget and plans, you will have to make compromises.

Right now your PC (i5 4th gen, 16 GB RAM, GT 730) is very weak for modern sims, but it can still run lighter titles like Assetto Corsa or older F1 games at low settings if you keep resolution and graphics down. For 200 to 250 dollars total, it is extremely hard to get a new “good” gaming PC. The realistic move is to keep your current PC and, if possible, look for a used graphics card later (something like a GTX 970, 1060 or RX 580) when you can save a bit more. That kind of upgrade will matter much more than buying a super cheap new mini PC that is not actually gaming capable. For a proper “Moza ecosystem plus modern sims at decent settings” experience, you will eventually want to spend more than 250 dollars on the PC alone.

On the cockpit side, a DIY wooden rig is actually a great idea for your situation. If you build it with solid 2x4 style lumber, screws and bracing, it can absolutely support stronger wheelbases and load cell pedals in the future. Think of it like a simple rectangle frame with a seat mount and a very solid pedal deck that does not flex. If you plan a bit ahead with measurements and mounting holes, it will handle a future Moza base just fine, and it will be far cheaper than most metal cockpits.

Your 32 inch TV will work. You do not need a monitor right now. Use the TV in game mode, plug in via HDMI, and you are good. A proper gaming monitor will give you lower latency and higher refresh rate, but that is a luxury upgrade for later, not something you must buy on this budget.