Hi! Sorry I was off in the woods and I'm still backlogged. Lemme grab my Master of Multisite Hat!
Database - You can cheat. go to yourdomain.com/dh_phpmyadmin/mysql.yourdomain.com and you can get in. Sometimes the alias breaks if you move servers. Open up a ticket and tell them that the mysql isn't redirecting to phpmyadmin Now that said, be very careful when you import your old DB over. You're on Multisite. Strange things happen on Multisite. If you don;t want to be on multisite, of course, you should read this: http://wpmututorials.com/how-to/how-to-disable-multisite/
blogs.dir - This is OLD. If you're on WP 3.5 or later, gleefully ignore that information. You can just delete it. It should be in wp-content, but I don't see that either so don't worry about it. If WORDPRESS told you to make it when activating multisite, that's different. Trust WP over the videos, always.
Weird Characters - This is due to your WP install being so out of date. When you upgraded, the wp-config.php was missing some of the newer calls (like define('DB_CHARSET', ''); ). I went ahead and rebuilt your wp-config.php using the wp-config-sample.php file (your old one is now wp-config-old.php), and just pasted in your sql and DB info. Yes, it really is that easy Characters back to normal! This is a very rare case, but it says you've been on WP since the 2.x days, and at that, I commend you! (I cannot suggest people 'fix' the DBs anymore because it's insane and complicated, and I'd just rather install a fresh WP install and do an import of the content at this point). There may be a couple outliers of weird content pasted in, but I did a spot check of pages and didn't see any.
Widgets - I see categories and archives... Is that what was missing?
And a moment of self-pimping, I wrote WordPress Multisite 101: https://store.halfelf.org/ebooks/wordpress-multisite-101/ (it has a sequel but don't worry about that), and while it's $7.98, there are discounts on the sidebar. Feel free to grab a copy and read it to get an idea of how and why all the moving parts of Multisite come together.