When the Shaliach Tzibbur is saying Kaddish, should he wait for the kehilla to finish Yehay Shemay Rabba ... and then say it after, or should he say it together with them, and continue with Yitbarach veyishtabach,,, ?

Shalom, Thank you for your question. There are various customs as to how the chazzan should say "Yehey Shemay …". In the work Piskay Tshuvot (56,2) he rules that the correct practice is for the chazzan to say the words together with the congregation, word by word. Unless your synagogue has an established practice otherwise, you should follow this ruling. Blessings.