Hospitals vary, but no, your DH can't shower, private rooms often full, and yes you will get food and drinks but none for DH (very rarely there is some spare). If DH wears a big jumper, it can roll up to make a pillow so he can nap by your bed in a chair . Some hospitals do now provide a more comfy chair to let dad rest better.
I couldn't get a private room for my post natal recovery, they were in use.
I had 3 small bags - one for before and during birth, one for baby, and one for myself after birth. Frankly you need a lot of stuff, and there is not enough room. Make sure DH is familiar with what is in each ag, so he doesn't have to rummage.
There is not much privacy on the induction wards, ime. Take headphones and have some films and music downloaded to drown out the noise of other people in pain (Hospital wifi is usually rubbish.) Long slow inductions can get noisy, I was left on the induction ward until I was 7.5cm dilated and had only had 2 paracetamol so I was getting noisy.
Is DH allowed to be there during the induction ? If yes then do include snacks for him, and check if he can visit your hospital shop ( hospital with DS2 had an M&S Food shop in it, fab).
For my first birth, hospital did have a toaster and a kettle in the post natal ward, with bread and spreads and tea available, only for mums to use. The idea was to get you up and walking around. In addition there was a tea trolley. Unless you had had a c section you were encouraged to make toast and tea for yourself. But for my second baby the hospital had no facilities like that.
Usually you choose food from a menu in advance. Spare food arises when a patient moves from one ward to another eg because they gave birth, then the food they ordered on the old ward isn't needed. It will be offered to new incoming patients first, so your first meal or two can be a bit random!
Typically at lunch and dinner there is a choice of soup, sandwiches, hot main meal, fruit, fruit juice and pudding, and a snack to save for later eg biscuits or crackers, you can select what you like. You get a jug of water that is frequently refilled. I had a supply of cereal bars and fruit in my bag as backup. Hospital food isn't amazing, but I always found the sandwiches okay.
Your hospital probably has an online video tour of the facility , most do due to covid - ask your HV. That will answer most questions.
Post natal wards are strange places. You can pull the curtain round for privacy. People try to be considerate ime, eg avoid noisy phone calls, but they are busy places. At night the other babies can disturb your precious sleep. The aim is generally to get discharged asap, as being there is exhausting, but use the time well to get help, in particular if you plan to breast feed, take every opportunity to ask a nurse to oversee your baby latching, there's a knack to it and once you get it right, it sets you up for success.
You are absolutely right to plan for a long stay. Inductions can be slow. My first was about 32 hours from arriving at hospital to giving birth. My second was quicker - just 16 hours.
I think the best advice I was given was, make your plans, and then be prepared to throw them out of the window. Planning helps you feel calm and prepared, but don't get too hung up on the plan. I wanted a fully natural birth for DD1, but i had an induction and there was a rush in maternity in the hospital so I ended up labouring in a regular labour room, got so tired I couldnt cope, had an epidural to help me over the line. Semt home next day. With second baby I had to be induced again, ugh, this time I did some hypnobirthing techniques and gave birth just on my 2 paracetamol and loads of gas and air (great stuff) - recovery was a lot easier for me but the baby had some difficulties so I was stuck in hospital getting the baby checked out for three miserable days, I could not WAIT to get home by the end of those three days!
Good luck, OP. Xx