A Takeout Double is appropriate for hands with a shortage in opponents bid suit(s) and shows support for unbid suits. Particularly unbid major suits. Hands that do not fit this pattern require extra values. A NT response to a T/O X is generally "unwelcome" and should express reasonable values (10?hcp). Try to have a 5 card suit for a possibly unwelcome minor suit response.
Rebids after a Takeout double also show extra values and are based on the minimum hand shown by partner's response. Consider that you have forced partner to bid and the response may be on a very poor hand.
1. East makes a TO Double. How should the bidding continue?
East's TO Double showed (if minimum) support for both major suits. With a minimum TO Double East should now pass (ie don't bid the same values again). It is now West's responsibility to continue the auction on the assumption that EW have a major suit fit. West was forced to bid and at this point has shown zero values. If East bids 2S here it shows extra values (a 6 loser hand).
2. East makes a TO Double and passes on the next round?
East has a minimum opening bid with support for unbid majors. If West's hand is such that he would have raised an opening major suit bid to the two level, it is now his responsibility to bid again. Unusually, holding both majors opposite a TO X and just enough points to bid at the 2 level, it is technically correct to bid four card suits DOWN the line. ie 1S followed by 2H on the next round gives partner an economical choice of contract.
3. East makes a TO Double and jumps on the next round?
East's jump to 3S shows a hand needing little support for West to make game and should be based on the worst hand West can have for the previous 1S response (ie a 3 card suit and zero hcp). West needs to appreciate that a spade honour and ruffing potential may be all that is needed to make game.
4. East makes a TO Double and rebids 1NT
East's 1NT rebid is based on the minimum values shown by West's response to the TO Double. ie zero hcp. The 1NT rebid shows 18+ hcp, too strong to overcall 1NT.