Congratulations, and welcome to the world of tubes! I think you'll enjoy it if your only audio experience has been SS gear. I can't answer all your questions since I don't have your specific amp and can only make broad generalizations about tubes. The best thing to do is exactly what you are doing: ask some questions and start rolling as you have the budget. Fortunately the tube compliment you have is not terribly pricey (as some amps can be). In addition to Andy at Vintage, you may also try John up in Canada at The Tube Store, for good and friendly advice and excellent service. Also you can do searches over on AudioAsylum in the "Tubes" section under any of your tube types to get more input than you ever needed. Finally do try to find other Avatar owners because different tubes have a different effect on different gear. OK, so here's some broad and sweeping generalizations for you to take with a bag of salt:
Chinese tubes tend to sound quite bright to my ears, although not nearly as bad as they are often made out to be. They are also very hearty tubes and tend to take more abuse and keep on performing. They are certainly considered entry level, and their sound is not very refined or warm in my experience. They are very cheap and manufacturers tend to include them as stock for that reason. You can do better.
The Russians do make some fine tubes, even today. In my experience they tend to be more prone to failures when new, but a reputible dealer would replace the tube if this happened. I've had at least three different Russian tubes fail in various amps over the past two years. The Sovtek's are probably considered the least desirable in most categories, though they have another higher grade of tube that is manufactured to higher specs under the name Electro-Harmonix which does manufacture some excellent tubes, among them their EL34's are outstanding in the current new offerings. The other EL34 I'd try is JJ Tesla, and if you can find the early 90's version of those in the blue and yellow box it is one of the very best made but they are quite pricey as NOS but not nearly as much as Mullards. Svetlana also make some good tubes. Their EL34's are nice but a bit on the colder side for my taste and in the gear I've used them in. Mullard's EL34's are considered of the best, but you will have to empty your pockets to buy a matched quad. I have no experience with them.
For your 12AX7 I'd go with Ei for current production, but if you really want to try an outstandign 12AX7 try out some NOS Tungsram's (I think The Tube Store has some of these). Pretty pricey at about $35 each but worth it to my ears and in my gear.
12AU7's I have no experience with so cannot tell you what I prefer, but ask around.
Of the brand names you mentioned as a question the NOS tubes by Mullard, Telefunken and Siemens are likely going to be the most expensive tubes available in many types and are all NOS as those companies are no longer manufacturing tubes. For NOS tubes, as a newbee, definitely buy your tubes from a reputible dealer like Andy, John or Kevin as they will stand behind their product. DO NOT try the eBay route as a newbee as you will be rolling the dice with most tube transactions there.
Hope that helps....have fun!
Marco