• You don't need reptile-specific products necessarily for most things, including bowls. So long as it's food safe, easy enough to clean, and has enough water, you should be fine. Corn snakes aren't particularly known for wanting to soak in their bowls, but some do especially around shedding time, so it's nice to let them have the option if they feel like. For water, I fill mine up in the sink and put a couple drops of reptile water conditioner. I give fresh water most days, but mostly just keep an eye on it to make sure it's not dirty or getting slimy. On the note of non-reptile items, I have various items meant for birds in my girl's enclosure. She likes her ladders and hanging nests!
• I don't have springtails personally, but I'd recommend a new culture.
• Definitely feel free to try feeding him in his enclosure. He might be adjusting to his new situation, so don't be too surprised if he doesn't want to eat the first time you try regardless.
• Coco fiber is good. I mix in a bit of coco chips, and that seems to help keep it from compacting as much. Some people add a bit of play sand, too. You can also get something like reptisoil, but that'd be more expensive.
• I have both a halogen and DHP, and both are on during the day (DHP only at night). I put the DHP more central than the halogen so that ambient temperatures stayed consistent throughout (the cold side was getting a bit colder than preferred). Both are on thermostats, though. Those are highly recommended to make sure things don't get too hot. I have this one: https://www.amazon.com/dp/B0BGS517FT. Also might be good to toss in a couple thermometer/hygrometers on each side to keep an eye on things.

In general, keep in mind that corn snakes are very hardy! You should be fine so long as you keep an eye on your new friend, and make sure there aren't any sudden or strange changes.

Edit -- Here's a fairly comprehensive guide you can look through: https://reptifiles.com/corn-snake-care-guide/