Unless they are managing updates/maintenance there is no such thing as a phpBB host. 2, 5 or even 10GB is relatively small amount of disk space by any standards, at least here in the US.
Thumbnails are built in feature on phpBB and have been going back to the orginal attachment mod in phpbb 2.X.We have got unlimited bandwidth, but it doesn't make sense to have pages grinding to a halt if a user puts a ton of massive pictures in a single topic. I'll have to try to find out how your thumbnails work.
ACP >> Posting Tab >> Attachment settings
Set dimension for the thumbnail and enable them, if you do not have a max image dimension that can be set on this page too. plupload will resize them client side if they exceed the dimensions.
This is getting into something that is complicated and something I'll be digging into myself in the future. In the header for HTTP response you can set how the image is handled for caching, the browser will cache it based on that but it can cached elsewhere. For example if you are using a service like Cloudflare they can cache it on their network and serve the cache version to the next visitor. This has two advantages, firstly the only resources consumed on your server is for the initial load. The second advantage is a service like Cloudlfare has a global network so someone halfway across the world will get much faster page loads because they are being served cached content from data center near them.I thought that image caching was just something that end-users had control over. Can image caching be set in the ACP? Or do I have to use something else (like .htaccess) to control image caching?
The reason it's complicated with phpBB is images can be in private forums, private messages etc. You need to make sure you are only allowing files to be cached that are viewable by guests.