Do I have to collect my luggage when transiting through Moscow?

On full-service airlines luggage will almost always, if not always, be checked through to the final destination if all of the sectors are on the same airline or on partners in the same alliance. This should definitely happen if travel is all on the same ticket which is not the case for you.

Some budget airlines do not check through baggage even when the journey is all on their network. Air Asia, for example, only do this on selected "FLY-THRU" routes and for an additional fee.

You appear not to have stated who the last flight is with nor if it is that flight which has the seperate ticket. Aeroflot may be willing and able to check your baggage through to Grand Cayman or they may not. If you need to know in advance you will have to ask them. I suspect your baggage will be checked through at least to Moscow but you can ask at the same time to be sure!

Of course at the very least this will all be revealed to you when you check in.

Make sure you leave enough time to pick up your bags and check back if necessary and confirm that you won't have to go through immigration to pick up your bags.

As far as I know you do not need to check-out, recheck-in your luggage.

I traveled from Austria over Moscow to Bangkok with AEROFLOT and vice versa and I did not need to look after my luggage.

In my case when traveling back from Bangkok over Moscow, AEROFLOT forgot my luggage in Moscow. I filled out some forms in Vienna and got my luggage 1-3 (I can't remember) days after I have landed.

Consider staying some days in the destination city (or in near the city) before traveling further, just in case to be sure.