Each instance of Minecraft initializes the mod separately, so they won't overlap ever. I can't answer to whether saving it statically is safe, but I'd assume so. However, you should only ever use Minecraft#getMinecraft() if you absolutely have to - there are almost always better and safer ways to get the information you want.
The player can only be gotten from a Minecraft instance on the client side, not on the server, so be very careful where you do this. If you can avoid using Minecraft#player at all, do. The relevant player will usually be passed along with events and method calls.
I can't answer to this for sure, but again, Minecraft#getMinecraft() is usually unnecessary and the same effects can be achieved in a different and safer way.
Hope that helps!