Depends who you are transferring to I guess. I know the IAA are currently dealing with a backlog and the process could take as long as 4 months. It is a fairly simple procedure and you can continue to operate as long as your current licence remains valid with appropriate ratings and medical etc.
As far as non-EASA aircraft goes you would have to talk to the authority you are transferring to and the CAA to confirm whether you can. As long as the aircraft doesn't require a non-EASA rating then I'm fairly confident that you can fly them with an EASA licence.
As for medicals you only need 1, as you can only hold one licence in any aircraft category (aside from national licences). IMC is non-transferable at present as the implementing rules regarding the EIR have not yet been formalised.