Could you clarify what you mean by "category"? Are you trying to move a topic from one forum to another at the same site, or are you trying to move your forum to a different host/server?
Which image are you intending to use? So long as you link the image to www.phpbb.com then I would say that it's OK, but I just want to check that you're allowed to use that image first...
On this page, the title is "phpBB.com :: Post a reply". On your website, which bit of that would you want to change - the site name (phpBB.com) or the page name (Post a reply)?
Could you explain exactly what is messed up? It looks OK to me.
Try clearing your cache and refreshing the page. If that doesn't help then try looking at the page in a different browser. Does it display correctly then?
The second problem is much more important than the first. It sounds like your host does not have PHP installed, which is required for phpBB. Who is your host?
Gah, a new version, 2.0.20, has just been released. You might as well follow the above instructions anyway to complete the upgrade to 2.0.19 first (just download phpBB 2.0.19 from here instead). But then after that you'll have to upgrade to 2.0.20. it seems your host updated your files for you last ...
Don't worry about the old backup. You can keep the database that was on your forum when the error started (assuming that's the data currently in your database, or that you have a backup of it). So, upload your most recent backup (or don't upload any backup at all if the database online at the moment...
Did you do anything to your forum just before this error started occurring? Because it is usually the result of an incomplete upgrade. Am I right in thinking that you have not uploaded a new version of phpBB recently? If so then your host must have changed the files without making the required datab...
As you said, this isn't the right place to ask questions about MODs. I see you've already asked at the site where the guide is. At phpbb.com the best place to post for support with this MOD would be its release topic: http://www.phpbb.com/phpBB/viewtopic.php?t=144548