Welp, the second part is clear; something is either old or out-of-date. I just checked a EE 2.5.5 install, and the exp_security_hashes table definitely doesn't have that column, nor any of those columns, in it's table conformation. In 2.5.5, that table looks like this:
Field - Type
------------------------------------
hash_id - int(10) unsigned
date - int(10) unsigned
session_id - varchar(40)
hash - varchar(40)
What is that third-party folder? /third-party/forms/? Is that DevDemon Forms? And did you update anything recently? If it was working before, it still should, but I have noticed some of DevDemon's stuff out of date, but that is usually only a concern if you upgraded EE. I would triple check the settings for that addon, as well as all of your site settings. Make sure you check in your /system/expressionengine/config/config.php file as well as your index.php file for any possible $config overrides that exists somewhere.
Another big trip up: what's your new environment like? Did you jump a bunch of subversions in PHP?