M.M.
Hi, sorry if this is direct, but you don't sound very experienced to me.
For the hours, it's time to renegotiate your contract! You SHOULD, as their employee, EXPECT them to keep track of how many hrs your are working. That's their job. You should be paid for what you're working, and I guarantee you that they are very aware of how much they are asking you to work. We have a nanny for our son, and we are hyper aware of when she works, and how much we are expected to pay her. And she lets us know when something is out of whack. She's our employee, we, her employer, and at the end of the day, that's how the "relationship" should work. It's professional, not personal.
The baby crying at that age is normal separation anxiety. It has nothing to do with you. All kids go through it. And it will get worse before it gets better! Your job is to distract the baby and comfort him. Make sure that he feels loved and cuddled, and try to divert his attention elsewhere - toy, cartoon, book, etc... They say that it's better (as a parent) to tell kids that you're leaving and will be back, rather than just sneaking out. So I would have mom kiss him goodbye and say she'll be home later, and then leave. When she hands him to you, you start the distractions.