I think the belt is going to wear excessively and may want to pull at idle.

Hard to be sure, but that looks like the power steering pressure hose.

It could be a couple of things. A parasitic draw or a bad replacement battery, or both. On simple test would be to disconnect the battery after parking it and after it sits for enough time to normally pull the battery down, then reconnect the battery and see if it starts normally. If it starts okay, then you have a parasitic draw. If it does not, you have a battery or charge system problem. It’s also worth noting that running the battery down multiple times is a good way to ruin a new/good battery.

Without doing any diagnosis, I would replace the brake booster vacuum check valve. It is most likely part of the hose/plastic pipe that plugs into the booster. In some cases you will find that the hard plastic pieces that get pressed into the plastic line, where the press together they crack and leak at those joints. Sometimes those can be sealed up with epoxy. One test you could do is check the effort it takes to press the pedal after you shut the engine off. In a proper operating system (not leaking vacuum), the booster retains enough vacuum to have a normal pedal feel for at least the first push. If the pedal is hard, then you would have a vacuum leak or bad check valve that is not keeping vacuum in the booster. In some cases it could be a bad booster, but that would be very uncommon… good luck, hope this helps.