Could we think of IMM dated swaps as forward swaps (since they trade only on specified dates and they might not be the current date)? For example, today is June 2nd, the next IMM swap is June 14 (not the spot date).
Broadly, yes. An IMM dated swap is usually just a standard swap starting on an IMM date. However, there are a few closely related instruments which you could be asking about:
IMM-anniversary swap - this not only starts on an IMM date, but also keeps to IMM dates for the roll schedule. It is a better match when hedging with IR futures, because an IMM-start swap would prefer Calendar anniversaries to IMM ones, and small date mismatches will occur.
Deliverable swap futures - in this case, the instrument is a future, not a swap, so it has all the complexity of pricing a future. The delivered swap, however, is a standard swap starting on the appropriate IMM date.
Note, by the way, that there are IMM dates in every month (third Wednesday), but usually it is the Mar/Jun/Sep/Dec cycle that is traded. If you need to hedge a May IMM swap, there are serial futures albeit they have nowhere near the liquidity of the major months.
Yes, that's exactly right. IMM swap are swaps that resets on IMM dates. Otherwise, the math is exactly the same as a standard swap.