If she slept from 7.30pm to 5.30am then that is an amazing amount of sleep for a 3 month old. I would have literally killed for that much sleep at 3 months. A baby "sleeping through" is defined as a 5 hour block e.g from 12am till 5am, so you can definitely say that she is sleeping through already.
In my opinion (not an expert or a health care professional), you can't teach a baby to self settle (except for via the "extinction" methods such as CC and CIO which I wouldn't want to do) - they will do it in their own time. In the meantime, you just have to do whatever to survive the sleep deprivation.
Half an hour in the pram is not a long time really. It would take at least that for my DS to drop off. I used to go on 1 to 2 hr walks around town until my DS was asleep. He would usually stay asleep in pram when I got home, so I could either nap or do things.
Wrt feeding, at 3 months I would just feed her on demand. If she woke at 5.30am I would just feed her whatever she wants rather than try and artificially stretch out or delay a proper feed. Especially if she has had a long sleep, as she is bound to be hungry. It can be as simple as "if your baby is hungry, feed them". I personally wouldn't try to push a baby into a routine, just see what pattern develops over time.
Some babies don't like shush pat, and some don't like pick up put down. I never let my DS cry for longer than a few seconds when he was under 6 months, and would go to him straight away. I was breastfeeding, so I would mostly just offer a feed, but if I was certain that it wasn't hunger, nappy, pain etc then I would pick him up and comfort him until calm. Then try putting him down again and seeing if he was sleepy. Shush pat means not picking them up and trying to settle them in their cot by shushing and rhythmically patting. My DS didn't ever really like that and would get more unsettled, hence me usually picking him up.