It sounds like you aren't in the DC area, OP. Is that right?
No TV is normal and healthy for those ages. Did they tell you that you cannot play music? Are they working from home?
You aren't stuck in the house if you can take the kids for walks. Do they have playgrounds nearby? A backyard?
Why do you need to clean and do laundry when the kids sleep? Do it when they're awake.
Maybe they don't have space for a swing or bouncer, or maybe they don't want one. It's their choice. Of course the baby will cry some when you put her in the pack and play.
How long have you worked for them? Plenty of families don't let you drive the first few months. Do they have a copy of your driving record? Have you gotten tickets/into accidents? Did you ask if you'll be able to drive the kids around after a few months?