At the start all I did was props and didn't really start sleep training till mine was 9wo and with #2 until she was 4months old. I think they only start learning how to self soothe at around 3 months and I wasn't too worried about accidental parenting because when they are so young and have been used to being rocked to sleep for 9 months, you have to give them some time to adjust to the new world.
I wouldn't do cio or cc... its just not fair to have your lo go through that. It doesn't work and you don't want him thinking that no one cares. Night sleep usually starts developing around 6 weeks so its great that he knows how to fall asleep at night and that you are having longer stretches. Day sleep at around 3-6 months. Some babies do find shushing annoying so maybe just try patting and whispering in the ear or humming. I did this with Jasmine - when I would shush, she used to get more hyped. But if what you are doing is working, that is enough! Honestly, 45mins naps is very common at this age. My lo only got over the 45 min naps at 5months. EASY might not look like true EASY for a while but as long as you stick to the feeding schedule & fit in naps in between, thats ok. If your lo is not falling back asleep for another cycle, pick him up and give him some more A time until he is showing tired signs. It could be in another 30mins and he'll probably have another 45min nap but this is very common behaviour. If its time to feed just before a nap, feed him... as long as he is not falling asleep by the bottle (or breast).
hth
Layla