oh Haddock, big hug. You poor thing, the early days and weeks are so hard.
Mine are now 11m and 2.10, and the early days are now (thankfully) a blur. It is so much easier now, and the beginnings of a relationship are so wonderful to see with my two.
The anticipation of your first trip out is always worse than the actual event. Yes, there will be times when it all goes completely, disastrously wrong, but you know, I'm a tougher person now than I was then as a result of it! You learn to brazen it out. I remember carrying a kicking, screaming DS1 across a carpark with a kicking, screaming DS2 in the Moby. Christ it was awful, but I didn't look at anyone and walked with a slightly manic smile on my face to the car and manhandled them both in.
There are days when everything is awful - you're tired, they're badly behaved and you just shout a lot. And then there are the days when everything goes smoothly and you feel brilliant and love them both madly - hang on to those days, and forgive yourself for the others!
In terms of bonding with #2, to my mind it's normal to feel like you'll never love number 2 like number 1. You know number one, you've been buddies for a while and he can chat to you and you have a laugh together. New babies are just bundles of need. You will get there and you will love this new baby madly too. It may take some time.
In the meantime, planning is your friend. Aim to get out in the morning and the afternoon as I find both DCs behave better when out. And are more tired as a result of it. Find baby groups, go to your local children's centre, go to the library, go to the park. Or just go to the shop to buy a pack of wipes. Whatever. But try to get something into each slot so the week doesn't feel insurmountable. Any interaction with adults is good.
Things will get easier with your oldest. I remember just feeling so unbelievably guilty that I'd wrecked his life. My babies have both been screamers (little darlings). He won't remember this time and before long DC2 will be his number 1 fan.
Speak to your HV about how you're feeling. They will have heard it before and they will not think you're an unfit mother. They will want to support you.
you are doing brilliantly. It is so hard. Another big hug.