Can I run a non-gfci outlet outside if it is downstream from a GFCI outlet on the same circuit?
Yes, you can do that, using the LOAD side of the GFCI.
But not only can you do it, it is actually a very good idea. That is because GFCI includes sensitive electronics not in a regular receptacle. Keeping a GFCI clean and dry and temperature controlled will likely make it last much longer than it would outside.
Sure you can.
If the downstream outlet is fed from the GFCI receptacle's LINE terminals, it will not have GFCI protection.
If it's fed from the GFCI recep's LOAD terminals, it will have GFCI protection.
Intending to protect downline loads is the only valid purpose of the LOAD terminals. They should not be used for any other purpose.