Sounds reasonably normal for private hospitals, but also a little light on the details and maybe the mum knows more and didn't explain in great detail or maybe she just didn't know stuff (like what the heart rate dropped to).
Maybe she was offered an internal earlier & declined. Maybe she even had one and didn't mention it to you. Maybe she'd asked for an epidural on arrival. Private hospitals are pretty happy to offer epidurals early in labour ... and maybe she was in agony on arrival & the anaesthetist was already there, etc. I've had internals before each of my epidurals just to make sure I wasn't fully dilated, so I thought that was normal, but I also know that the more internals you have the greater risk of infection, so maybe they decided to do the epidural without an internal for some reason.
My dd had formula as a top up for the first 2 - 3 days, but she was big (4.4kg) and hungry (aka starving) and had low blood sugars (I had gestational diabetes). Just cause your friend had big boobs didn't mean she had heaps of milk on day 1 & 2, virtually no-one has heaps of milk at that point. Colostrum yes and the stuff is liquid gold (very high in energy), but it's not big in volume at that point. Your friend said the formula was so she could sleep, but that is what she told you, but was it cause she'd been BFing constantly for 6 hours and was in agony from her c-section scar and in need of a couple of hours of rest? Remember she just had a c-section and may not have been even able to get out of bed the first night, so going to the night nursery is a safe option and it wouldn't have been for the whole night at all.
My dd went to the night nursery for a portion of every night. The first night was cause I needed sleep and she was coughing up sputum so they wanted to watch her in case she needed suctioning (she didn't). The other nights she was SUPER unsettled (she had a broken collar bone) and nothing we tried helped her settle for long with me and I was exhausted. So, she spent maybe 3 - 4 hours in the nursery. To me, I would have been better at home cause dh would have had her in the lounge room & I would have got sleep, so comments about 'get used to a baby that cries' are really not helpful.