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1) Would you open 1NT instead of 1♣?
Yes, but I accept this is not the norm with this distribution, and even some of those who do open 1NT don't do it with small doubleton.
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2) Playing StrongJumpShifts had you bid 2D as first response?
No. I play Soloway (or is it Goldman) jumpshifts, showing suit that can play in slam opposite small stiff, or good suit with support for partner. I lack either of these requirements. And I don't play strong jump shifts anyway.
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3) What is your rebid after 1D?
1S, what is the problem? This promises an unbalanced hand, with clubs and spades. So I ahve a club more and a spade less. I have three really good spades. If partner has four spades, he has a good hand for his 1D response. 1S also allows partner to use XYZ (or fourth suit forcing). Also note, my system specially allows rebids of 1H or 1S on three card suits after 1C-1D as my 1NT rebid here shows 17-19 hcp, but this is not standard. But evne not playing that way, 1S has to be right.
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4) I'm sure I should bid 4D after 3NT, but how to find then the grand?
Ok, staring with your first two choices..
1C - 1D
1S - 3D (where 2D is GF xyz, but 3D is also GF but more distributional hand)
4D - 4H (4H here is kickback)
5C - ?
(so all five keys held, what else do you know? Partner lacks a heart stopper and is not balanced. In addition, partner bid three suits, so likely singleton heart (see why opening 1NT is better?). No need to ask for heart king, but you can ask in spades with a specific suit asking bid (see kantar's RKB book)... which is 5S. Partenr will raise to 6S showing KQ. Now you count your tricks, 7D, 3S, 1H, 2C (partner did not bid 3C with Queen empty suit, surely he has the king).
----- 5S
6S - 7NT
Pass
Ok, that worked well, and easily. What if you had opened 1NT?
1NT - 2NT (diamonds, weak or strong, or weak minor 2 suiter)
3D - 3S (3C = I like diamonds --bidding same if you bid 3C don't like them)
3NT - 4D (3S weak spades, 4D = Keycard blackwood now, GF)
4NT - 5NT (4NT = 2 keycards, 5NT = specific king ask)
7D - Pass (5NT promised all key cards, I have K-sxth is clubs, and AKQ of spades, these have to be enough for good play for 7.
At matchpoint 7NT would be bid.