Hi electrichummingbird,
Do you have a custody agreement in place with your ex? Skimming your posts, it sounds like you proposed the current Skype schedule, and not the court?
If the Skyping is court-ordered, that's a different email. If this is something you proposed, then it's really about boundaries.
Boundaries are really easy when you don't feel fear, obligation, or guilt (FOG)

When you feel obligated, which it sounds like you do, then boundaries seem very difficult and onerous. An email to your ex could be as simple as:
Going forward, the new Skype schedule will be Monday, Thurs, Sun, between 7-7:30pm. I will make D2 available between 7:00-7:10pm -- if there is no call between those times, I will assume you won't be calling that night.
We often recommend keep email communications short and to the point.
Skyping twice a day for a 2 year old is likely to be considered excessive by anyone affiliated with the court. Skyping at all with a 2 year old is probably going to surprise people for exactly the issues you have to contend with -- like running around with the laptop.
When I first left my ex (before temporary orders were in place), my son (9 at the time) talked to his dad on the phone every day. When we went into mediation, everyone felt that was way too much, including my ex (!), who had asked for that schedule.
Over the past 3.5 years, there have been two long stretches when visitation was suspended, and no one -- not the judge, not the lawyers, not the therapists, not the parenting coordinator -- thought it made sense to have daily contact. When we drastically cut visitation between n/BPDx and my son, they tried Skyping but S12 often had nothing to say and their calls sometimes lasted less than 30 seconds. Eventually, N/BPDx just stopped calling. Fortunately, the custody order said he had to initiate the call, otherwise N/BPDx would've used it to try and control me. Even so, I resented having to rush back three times a week at 6pm to get S12 in front of the computer. It made afterschool activities hard to plan and during the summer we often had to interrupt plans because there was no way to reason with N/BPDx about that kind of thing.
Also, if you are initiating the calls, it's best to put the burden on your ex to make the calls. A lot of times, especially in the early stages of divorce, and especially if you have a BPD or NPD parent, setting boundaries is going to feel really mean. Setting boundaries, though, is about taking care of yourself. A lot of times, we feel guilty about that, but once you work through those issues, setting boundaries feels a lot healthier.
If you need to do this in small steps, figure out what the guilt is about. If it's about wanting your ex to have a constant presence in his son's life, then tell him he can record things and you'll play them for your son when he has some downtime. If you feel guilty about your ex not seeing his son grow up, then take lots of pictures and videos and send them to your ex at your convenience.
Another issue here -- this is about your ex putting his son to bed at night -- is that whatever you do now establishes status quo when things get finalized in court. If you don't think it's reasonable for someone recovering from a major operation to put your D to bed, then it isn't reasonable. The court will find you to be clear-thinking and reasonable. If your ex is in no shape to care for your D at night, and you hand her over, then the court doubts whether you are a stable parent.
Last -- about your SO. It seems early for a new partner to be advising you about what you should or shouldn't do. He's probably correct that there is something manipulative about the daily contact, but I would guess it's as much about your ex controlling you as it is wanting to reconcile.
If you don't have a custody order, you aren't stuck doing anything. You don't have to do anything you hate, or agree to anything.