If you want the cheapest option, minimum I would recommend is the TS18. It might not be as powerful as the TS10/7832, but it's good enough (I am also using one).

Everybody is still waiting for new stocks it seems. The combination of people buying a lot of head units after getting the 10K EPF (pasang sebelum balik raya) and chip shortage means that a number of sellers don't have new stocks right now
Same reason as above. I also am surprised at how hard it is to find it right now. During the first week of Puasa, there were a lot of shopee sellers selling this around RM550++. Nowadays not many have stocks

Btw your Kenwood link is suspicious. I doubt they are selling a "real" 4GB unit below RM300. Likely it's a modified firmware that tells you it's 4GB when it's only 1GB

Both shop is legit, so my take on this is choose with your wallet (RM590 vs RM745)

Yes, it's similar to how a normal radio work.
But you can also set that the device only turn off it's screen (similar to how you lock your screen of your phone, for a max of 3 days) when the engine is turned off. If you are a daily user of your car, then this should be fine and should not drain your battery much