Re: What is a typical day for a 2 month old?
I recall that our baby slept a lot like you are describing in the earlier days. I don't recall exactly when she started staying awake more in the day, but it seems it was somewhere between 2 and 3 months old for us. Also, at 2 months old, our baby was very colicky, so when she was awake, she was screaming. So we didn't have much play time. I held her a lot to constantly try to soothe her.
About 3 months old, the colic stopped, and I'm guessing around 3½ months old, the naps stopped completely, and now at 4½ months, she doesn't nap at all...not even cat-napping anymore. We have a lot more play & interaction time now that she's alert & aware of her surroundings.
As for falling asleep in your arms...I loved that!...naps and bedtime...and I miss it! In hindsight, letting her fall asleep in my arms at bedtime was something I should have put a stop to earlier than 4 months old...and let her learn to go to sleep on her own. I'm talking about bedtime here.
If I could do it over again, I would have started putting her to bed while she was still awake & aware at around the 3 month point (when the colic stopped) so she could have learned to put herself to sleep sooner. At 4½ months old (actually, closer to 5 months now), we have come a long way with bedtime...letting her CIO, but still have struggles with getting her to go to sleep without wanting to be rocked to sleep. I'm just hoping she learns to put herself to sleep before she learns to say "Mommy". It would kill me to have to hear her screaming "Mommy" from her crib.