We recently moved here from SE London with our three children and are renting until we can afford to buy again. At first I thought the same as you - St Albans would be nicer but was out of our price range. But having lived here 8 months I think I actually prefer Hemel for our family at least. The traffic is less hectic for starters ;-)
Anyway, in answer to your questions:
a) Boxmoor, Chaulden Vale (not necessarily Chaulden itself), Leverstock Green, although it's annoyingly far from the station, parts of Apsley (the marina, Manor Estate, although the latter suffers from train/A41 noise) are all nice.. obviously you pay for this though. The really nice parts are in Felden, super expensive though. The easy way to tell if a place is not that great is what the properties are going for - if it's cheap, it's not as nice. Having said that, compared to where we lived in London, any part of Hemel is far, far nicer. We made up our minds by getting shortlists from having a look on Streetview, then doing drive-arounds to get a feel for the place.
b) The best primaries are considered to be South Hill (Boxmoor) and Two Waters (Apsley), plus there's a good academy at Bovingdon I think, and I think there might be a good academy over at Woodhall Farm. Boxmoor School used to be one of the best, it is a good school that suffered a few years back from a retiring disinterested head, but it has turned around again now and is most definitely on the way back up again and has always had good SATS. Micklem (Chaulden) is also a school rapidly improving thanks to a fantastic new head. St Roses's is a good catholic school. There may be more, but those were the ones that kept cropping up during my research.
The Hemel School is a decent secondary for the future, and it's not impossible with the right scores (11+ etc) to get into selective schools in Watford or Chesham - there are direct buses.
I didn't do a lot of digging about with nurseries because we were happy with the first one I looked around - Positive Steps in Apsley. Very reasonable price, and very reasonable and lovely management plus the staff there are great too. I've heard negative anecdotes about Lime Hill (leaving children to cry) and Flowerpots (apparently the staff were like air hostesses!) but these are just stories reported by one parent each time so aren't necessarily indicative of a place.
There are loads of playgroups for toddlers. I never took mine to any as they were on the wrong days or in the wrong places, but there are plenty about. The gymnastics place does a soft play every weekday morning for 45 minutes which is popular. I think the library does some reading stuff on some weekdays too.
I love the place. Sure, the town centre is, well, a bit shit, but it's no worse than most towns of this size that aren't "chic" (for want of a better word) like Berko for instance. The usual chainstores are there, the TK Maxx is pretty good - you can buy what you need to in most cases and there's always Milton Keynes or Watford if not.
Town Centre aside though, it's perfect. We're an active family so we really appreciate Sportspace with its five swimming pools (the outdoor one is fabulous in this heat!), the one-of-the-best-nationally gymnastics club, the Snow Centre with real snow, the climbing wall (massive!) - all of these have classes for 4 years old and above, and in some cases younger. I think in most cases there is nothing better for these activities for quite some distance. Plus never mind the masses of walks/runs along the canal, through the forests etc. And 25 minutes into Euston is not to be sniffed at, and 20 minutes to Luton airport (yet not on the flightpath) is incredibly convenient.
I've found the people to be very welcoming, chatty, friendly and possibly a bit more diverse (racially and culturally) than some of the more expensive towns locally. I have never, ever, once felt unsafe or intimidated, even after catching the last train back from London and walking home in the pitch dark on my own.
And some parts are very beautiful - the actual moor at Boxmoor with the wild ponies (loads of foals right now), the canal, the woods, the Nickey Line disused railway line which goes to Harpenden.
If we want the nicer shops/restaurants etc of St Albans or Berko, we drive or catch the train, so I don't feel we're missing out. They're still there nearby for us to enjoy. On a daily basis I use things like the various fabulous sports facilities far more often than I feel the need to go shopping in smart, independent shops, so on a practical basis it's a better location. There are a lot of parents at the various activities we do that have driven in from other towns and been fighting with the traffic.. in most cases we have just walked over. I read somewhere once that although it is lovely living in a town with a (say) nice cathedral, how often would you actually visit it? For me it's about the practicalities!
I really feel we hit the jackpot living here in terms of suitability for our family (everyone is different after all with different priorities) and have no regrets! :-)
Hope that helps a bit! Sorry for the essay, but I figure the more the merrier plus I never know when to stop!!