6:30am is not an unreasonable time to wake up. What time does she go to bed? If she's going to bed around 7pm, then she's getting a good amount of night sleep. Both my kids (unfortunately) tend to wake up around that time so I think it's pretty normal.
Since she is getting older now and moving beyond the newborn stage, I would try to get her on a regular eat, activity, sleep pattern when she wakes, so it would look something like this:
Wake: 6:30am
E: 6:40am
A: 7am
S: 8am (hopefully 1 1/2 hrs but she may be a short napper as many babies this age are)
E: 9:30am
A: 9:45am
S: 11:00am
this pattern will continue through the day until bedtime
Don't be concerned about APOPing her to get her to sleep at this age. She's just learning to sleep and you have lots of time to train her to nap in her crib in the months ahead. It's just really important to get her the sleep she needs so she doesn't get overtired. My DS slept mostly in the sling for about a month because he would only do 30 or 40 min naps in the bassinet. When he was a newborn, I always just tried to do at least one bassinet nap a day and then tried to extend it by rocking him back to sleep or holding him. At about 3 months I started really crib training him and trying techniques like wake to sleep and holding him through the transition jolts.
In regards to swaddling, I would definitely keep trying to do the arms in because she'll probably fuss herself awake even if she falls asleep. She just doesn't have control over her arms yet. Sometimes when they fuss as they are being swaddled, they aren't actually objecting to having their arms down, they are trying to calm themselves. See if she will calm down after a few minutes after she is swaddled as you are holding her.
I would definitely do a similar but obviously shorter wind down routine for naps so it cues her that it's sleep time. Keep the activity just before nap time very low key. Babies are way too stimulated sometimes to fall asleep especially if she has lots of dangling toys, activity in the house and visitors over.
Good luck!