When looking for PHP classes, I rely on searching phpclasses.org and evaluating the many classes offered (free) albeit it may take some playing with several before finding the best fit.
As for your requirements:
1and1 has a reputation as a terrible host with a severe lack of services (.htaccess, full cPanel, non-responsive or slow servic and the like if I recall correctly) so I'd advise finding a better host (I have multiple accounts with WebHostingBuzz and find that they are outstanding).
E-mail verification is simply a matter of using a two step registration: Sign-up online then respond to an e-mail generated by the server and sent ONE TIME to the registered e-mail address.
Member Group Assignment is a feature requiring moderation (like a forum). I'd developed such a feature for a club and it worked well ... except for the President who believed it was his right to edit everything any of the committee chairmen did online - then wanted to retain that capability after his term expired (that's the problem with volunteer organizations like this).
Other than the 1&1 comment, I'd recommend refining your requirements, i.e., determining whether you're trying to build a forum (most forum apps I've seen are commercial apps), discussion group (MailMan seems a good option for this and is free) or something else.