Congrats on the good parental status! I hope it continues.

Rushing is never a good idea, but you should move at the pace you're comfortable with. My wife thought I was rushing, but I thought the complete opposite, lol. There's just a lot of processes to start the balls rolling on, and I felt I needed to get them all going at about the same time.
1) That's a weird comment to hear from a laser hair removal place. My laser consult actually said, "your hair may actually get thicker/darker with HRT" and I thought that was stupid too though. You should start laser as soon as possible though if you hair/skin color is suitable to using a laser in the first place. My beard is almost entirely blonde now and it's only been two sessions and 3 months on HRT! Unfortunately that means I'm probably going to need a lot of electrolysis, but the brown hairs are nearly gone. Start both as soon as you're able to and are comfortable with the changes that will ensue.
2) I didn't save sperm, but my wife and I got pregnant 4 months before coming out as trans. I really can't comment on this question. Sorry! I don't plan to bank sperm moving forward either.
3) Yes, you need some girl time! If you have friends that will help you, that's a big boon. You can visit the LGBT center, they have some support groups I'm sure that will help you feel comfortable being a girl in public. Go shopping, a lot. It gets easier with time. You eventually stop caring that you're carrying girl clothes into the girl's changing rooms and you're just there trying to figure out which cute top to buy!

Go get your nails done, and/or your eyebrows, start growing out your hair. If you wear glasses, get some feminine frames. Find a nice purse you can carry too because if you start wearing girl pants, you won't have room in your pockets for anything! Oh! And find a good bra with cups that hold their own shape, until you can fill it out yourself. Just wearing a modest size cup bra and a t-shirt is so reaffirming to my sense of self.
Hopefully you're not too young to do some of these things, not sure how old you are, but they all helped me.

Going dress shopping with a girl friend of mine was really helpful, and she also would go into the ladies room with me, hehe. I'm so much more confident now, but it does take some help to get started.