1. A hybrid or mountain bike with slick tires. Add fenders, racks, and lights and you're in business.
2. A trailer you can leave at school is easy. Chariot double or Burley D'Lite (this is the most high end Burley) will last you a while - both have suspension and fit 2 average size 5 year olds. The Burley D'Lite has more space and the seats collapse to make a large cargo area; the Chariot has a nicer easier to use hitch and is easier to find used. You need a safe storage place though -- trailers are impossible to lock securely without disassembling them.
If a trailer doesn't suit you, a Yepp or BoBike baby seat will work for a while and when your kid is older you can switch to a junior seat, a trailer bike, or your kid riding her own bike (my 4 year old rides a mile in about 10-15 min).
3. Flags and lights. Don't skimp. I have a flag, some battery operated waterproof LED Christmas lights all over the trailer, monkey lights on the wheels, and 2 cygolight flashers on the handlebar. The thing is lit up -- I get comments all the time on how well lit it is. It's so outlandish, it makes people smile, so they're less aggressive and don't try to kill me (and my kids) as much.
Sounds like you need an electric mountain bike too ;-)