You can obtain the TM Sunny Day from the Radio Tower located in Goldenrod City.
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Mimic is not a TM. In HG/SS, there is also no Move Tutor that teaches Mimic.
If you still have the TM/HM, teach it to the Pokemon again. If not, go to the move deleter-- there should be a move tutor there too. I learned through HG/SS though. |D
Professor Oak's assitant will give it to you. Oak's assistant WON'T give it to you. He does that in RBY/FR/LG. In Pokemon Gold, Silver, Crystal, HeartGold and SoulSilver, the Elder in Sprout Tower will give it to you. In GSC, Flash is a HM, in HG/SS it is a TM.
In HG/SS you can have tea with her any day from 3:01 - 4 pm :)
You cannot trade a Pokemon from the 5th generation to the 4th generation or below. Getting Pokemon from Black/White to HG/SS is impossible. You can migrate Pokemon from HG/SS to Black/White though.