Just undone the wrong screws on the calliper and a bit of fluid came out. Quickly did them up again. Now the lever goes very close to the handle bar before the brakes kick in – they don't feel spongy though and do stop well. Do I need to bleed or is there some other adjustment? (I don't think I can add fluid in the top reservoir like my hopes)

better safe than sorry i'd bleed them just to be sure.

Do I definately need a Avid kit for that or it it possible to use a home brew kit?

you can make your own kit if you like, sourcing the syringes and oil elsewhere but the screw in adaptors for the reservoir and caliper ends might be a little tricky.