There is hosts that specialize in supporting some web application like Wordpress but I'm not sure about phpBB. Basic hosting packages do not include support for web applications and ones that do aren't going to be cheap especially if you have customized styles, extensions etc. Each customers installation would need to be treated on individual basis. Be wary of what you are purchasing.
Check the signatures here of staff and long time members. Many provide paid support and you can contact them about paid support if they have it in their signature, they are very familiar with phpBB and it might be better path .... and no I don't do this myself.
There is also the wanted section:
viewforum.php?f=656
Be aware we do not monitor people offering services there, buyer beware.
2 - Also how should I go about to ensure nothing is lost when transferring all this data?
The easiest thing to do is buy one month on new host. Maintain your current plan with existing host. Once the forum is set up on new host you (or whoever) will change the DNS entries to point to new host. If you need data or something is missing all your files and data still exist on old host. Worse case if you decide this was very poor decision and new host sucks you can switch DNS back to old host, you are back where you started.
Once everything is straightened out and if you are satisfied with new host cancel plan on old host. As side note domain registration is typically included with hosting plan but it really should be on separate plan provided by some other company. Specifically for cases like this and other reasons, domain registration with decent registrar is only about $15 annually.
“Results! Why, man, I have gotten a lot of results! I have found several thousand things that won’t work.”
Attributed - Thomas Edison