Q6: +1 card token on Throne Room (with Teacher). Play Throne Room, Throne Room, Throne Room. The 3rd Throne Room doesn't play an action. The 2nd play of the 2nd Throne Room draws and plays an action card that draws cards.
Q7: Venture
Q8: The other action is also a Rats
Q9: +1 card token on Throne Room (with Teacher). Play Throne Room, Throne Room, Tactician. The 2nd play of Throne Room draws a card, that you can play with the Throne Room.
Q10: I don't know. You play something afterwards to draw the discarded card?