Newbie to Sofia,info needed please

Peppi,
You don’t provide enough details to be able to advice in any meaningful way.
There are lots of nice areas nearby, geo milev, iztok or even studenski grade if you like.
It really depends on taste and preference.
I am a bit puzzled why it needs to be walking distance though. Sofia transport network is excellent and very cheap by UK standards. You buy a talon of tickets, 10 single journeys as far as it goes. Try this in London, all for less than a fiver.
As with reasonable priced apartment, reasonable to whom, someone on 2000 lv 3k or 5k, how long is a piece of string.
My advice would be as many do, go there a few days earlier,book a cheap b&b and spend a few days finding accommodation.
Bulgaria also has a health service and although it may be way behind on the equipment level it is not a overloaded as in the UK, many of them are probably here with us in Western Europe, I know some of my Bugarian friends are.
The employer may provide some element of healthcare, but in any case, cost is hugely dependant on service levels and of course age and provider.

Wouldn't recoment studentski grad for a family, otherwise a 2bdr in that area shouldn't cost you more than 500-600lv a month furnished. Again you can find it cheaper and for a lot more. Prices tend to go higher if you are foreigner but just negotiate and have at least three flats that you like so you can just walk away if the price is not right.