Extras are probably (though I'm not familiar with your tariff so someone else might know more....)
-Internet use as you mentioned. May be capped at £1 per day (or less if you use less), I'd check with O2 to be sure.
-I think MMS are extra on that tariff.
Wireless at home will be free.
I assume your Nokia Maps has the maps pre-loaded on the phone? If so it will connect via the internet for a very short time so that the A-GPS can help you get a lock on. You can disable this but it will take the GPS on it's own longer to lock on. If there is no GPS on the handset it may try to connect to give you a rough location, but this should only be a few kb of data. You should be able to turn it off in the options too. Google Maps will cost the earth though as it downloads the maps as you go.
I check usage by going into O2 Active / Mobile Internet on my handset, then MyO2, then MyO2 Account. If you bookmark it it's then free to check. It can be up to 48 hours behind, but mine is generally only a couple of hours. You can also send a blank text to 21202 and you'll get a text back with your info.
No idea how to transfer contacts. If both phones are O2 (or the old one is unlocked from any network) you can copy the contacts onto the sim, then onto the new handset.