Looking for recommendations on a cheap/budget racing wheel and a game

if you are a little handy and know a little bit of electronics you can make it yourself for a cheap price

I'd personally stay away from cheap wheels. The ones that have "vibrations" are nowhere near as good or "realistic" feeling as the force feedback ones. I'd look for one with >270 degrees of rotation. Also, look for the ones that have clamps as a mounting mechanism, the ones with suction cups don't usually stay in place. I'd buy a used one that's in good condition, although it can be a gamble. Nothing worse than getting a banged up wheel. Options like a Thrustmaster T300 (RS)/TMX (Pro)/T150, Logitech G920/29 probably won't fit your budget, but maybe you can find a Logitech G27/25, or even a trusty old Logitech Driving Force GT for cheap.

As for games, you'll find free roam in something like Forza Horizon 3/4, The Crew 1/2, Need for Speed 2015 and Payback, maybe GTA V, but that's about it for newer releases. Older titles like Test Drive Unlimited 1(with the Platinum mod) and TDU 2 are really fun. FUEL also has free roam I believe and is set in a post apocalyptic setting, but I'm not sure it benefits much from the use of a steering wheel. Dirt 3/4 and Gravel are fun for casual rally racing. Wreckfest is fun for demolition derby, as you know. If you wanna give track racing a go, maybe try Forza Motorsport 7, or GRID, or NFS Shift Unleashed. If you like driving trucks, Euro Truck Simulator 2 can be a fun experience. There are also F1 2010-2019 for your F1 needs, but they're not exactly casual. If you want to get more serious about it, there is Project Cars 1/2, Assetto Corsa, iRacing, RaceRoom, rFactor 2, Automobilista and other specialized sims.

I have the Thrustmaster Ferrari 458 wheel and I like it. It's available for 80 USD here in the US, but I don't know if it's available where you are.

"Free roam + racing when you want" is not too easy to come by. I'd echo PrototypeOnyx and recommend The Crew and GTA V. Both games make it fun to just drive around randomly.

If you're on a budget I'd suggest taking a look at your local classified ads websites and search for a used one that is in good conditions.

Lots of people buy expensive wheels, use them a couple times, then get bored and sell them.

Logitech Driving Force GT, 900 degrees wheel radius, buy used, got it for 60€.