Answers: (More advanced material is in blue.)
1.
Hint: What do you think your best contract is here? Do transfers make any sense in this sequence?
Answer: Your partner is showing 19-21 points and a balanced hand. If your partner opened 1NT or 2NT, you would transfer to hearts and sign off. You want to sign off here also, as your hearts will not only take tricks as trumps, they will give you entries to finesse against the opener's high cards.
As you have already bid hearts here, there is no reason to transfer, and 2D would show a diamond suit rather than a transfer. The recommended call is 2H. You aren't showing any strength with this call (except that you're saying there isn't enough for game); you are simply placing the contract.
2.
Hint: Do you have enough for a game? Do you know the right place to play?
Answer: Your partner's 19-21 plus your 7 points should be enough for a game. As you don't have a major suit fit (partner would have raised your hearts with 4-card support), just bid 3NT.
3.
Hint: Do you have enough for a game? Do you know the right place to play?
Answer: If your partner has a balanced hand, you have an eight-card heart fit. Again, while your hand may not have the entries to make your hearts good in notrump, you want hearts to be trump to make sure you will score some heart tricks, as well as provide entries to finesse against opener.
Since your six-card suit does have some value, you have enough strength to make game. When you know where and you know how high, just bid it. The recommended call is 4H.
4.
Hint: Do you have a game? Is it possible to give partner a choice of contracts?
Answer:
You have enough strength for a game with your 7 points (8 counting length) opposite 19-21. You would like to play in hearts if partner has three-card support but play in notrump if partner has only two hearts. Can you invite with 3H?
No, because you could have a hand like Hand 1 which wants to sign off in 3H. Partner will pass 3H.
Is there a bid that would give partner a choice? Let's think for a minute.
Let's pretend that when your partner doubled, you knew you had a game but you didn't know which one (let's say you had 4 spades and 4 hearts and 13 points.) What would you do since no new suit bid is forcing? You would cue-bid. It would show at least an invitational hand and doubt about what, if anything, should be trump.
You can cue-bid now to mean the same thing; I believe there is a game but I don't know which one. Since bidding 3S will virtually force your side to game, you can't use it on an invitational hand anymore, but you can tell your partner that your side has the strength for game but you don't know which one.
He should recognize the problem. You don't know if there is a heart fit (so you must have five) and you want him to choose between 3NT and 4H.
If you are new enough that you are unaware a cue-bid exists, take full credit for either 3NT or 4H.