I agree w/everything you stated about working in a home w/a parent who is in the home. Yes, the child DOES act much different knowing their parent is in the home and yes it is very stressful when you have to keep that child from bothering the parent working in the home office. BTDT.
As for a mother on bed rest, I am not sure if this job will be any different I am afraid. I would only accept this type of job if I really needed the money because I personally feel uncomfortable working alongside another parent due to micromanaging issues. I prefer to work independently and have complete autonomy at all times, but that is just me.
If you think you can make this work, then yes, I agree...It is important for you to ask the mother if you will be able to take her child on daily outings while in your care. If she is not okay w/that, then I don't think this job will be a good fit. To have to stay indoors w/both the child + the mother all day would drive me stir crazy!!
Also, this may be tricky, but I would want to know exactly where the boundary lines will be drawn. How will the day be structured? Will you be able to plan the child's daily activities? Or will the mother plan them and you will simply follow her wishes? During the child's nap times, will you be expected to do any non-child related household duties? And if so, which ones? Are these acceptable to you?
All of these things must be pre-discussed before hire.
Hope this all helps. Good luck.