thanks for taking the time to post this bug report! I'll try answer what you can do in some cases, or why what seems strange is actually correct.
1. The card selection seems to always pick the same cards (Margrave, 5 games in a row!)
Make sure that either you make a table that doesn't respect familiar cards (you can adjust that in the Advancved Options of a table), or make sure all players have enough different cards marked as familiar.
Familiar Cards can be changed in the "Familiar Cards" tab, roughly in the middle of the top row of buttons in the lobby.
For a drastic measure, just have you and each of your friends click the button "All expansions" in the "Familiar Cards" tab.
2. The in-game algorithm is wrong:
2.1 Round 3, you always draw whichever card you purchased on round 2 (in fact, any time your discard pile is empty you will draw the last card you gained)
You seem to be either extrapolating from a very small number of occurances or having some extreme shuffle luck. Just my first test game I draw the card I purchased turn 1, for example.
2.2 Literally draw the exact same cards in the exact same order multiple times a game
If you buy multiples of a few cards, you are bound to draw similar sets of hands. And what are you basing the "exact same order" on? The order of cards in your hand and what is written in the log is *not* necesserily the order they were drawn.
2.3 If there is not enough cards in the deck to draw, it will shuffle the discard pile and deck together before drawing, meaning stacking the deck is not possible in some circumstances
That's a new rule about drawing multiple cards since AFAIK version 2 of the base set (I may be wrong on the exact time that went into effect). If you are supposed to draw (or otherwise "handle") more cards than are in your deck, you shuffle the discard pile and put it under
the deck, then draw.
It's supposed to avoid confusion when you draw for example for Hunting Grounds (+4 Cards), realize in the middle that your deck is up, then you (under previous rules) would shuffle your discard pile and put it down as your deck, then wondering how many cards you had already drawn. Under the new rule, you draw them all together once you realize there aren't 4 cards and you did the "shuffle and put under".
And that is exactly what happens in the online game as well, the discard pile is shuffled and put under
the remaining cards of the deck.
Each game has game number, in the 39 millions around the time of posting. You can find it at the very top of the log, or in the chat (green, "Joining game 39XYZ... on Tokyo/Frankfurt/...", make sure to look for that last such entry). If you think something went wrong in a game, note down the game number and post it here, with a description of what you think went wrong. Then I or others will look at the game and can see if that is really an error or not. Lots of bugs have been found this way!
Again, thanks for reporting this.