Can you have both 15 and 20 Amp receptacles off the same 20 Amp GFCI?
Yes you can.  



(I see the OP was edited to include details and will retain the rest for the database)

Having the GFCI marked 20 Amp is so the one model can be used for either type of circuit.  If you have a 20 Amp Circuit then there is no problem with using 15 Amp receptacles on the same run.

BUT

Having the receptacles is one thing.  Having the wiring and circuit breaker to support 20 Amps is another matter.

Unless you have the 20 Amp breaker and proper wiring, you will want to limit the total current draw to 15 Amps.

PLUS,

Most 20 Amp circuits are dedicated for specific purposes.   Most often there will only be one single outlet for a specific need like a computer rack or a dialysis machine.  

In a GFCI situation like described here, the second outlet would most likely be very near to prevent causing the breaker to trip inadvertently.  Example:  Plugging in an electric heater.  Something that has nothing to do with the original reason a 20 Amp circuit was installed.



ALL THAT BEING SAID,

Yes, you can have either or both kinds of outlets on the same circuit, and it will work well unless you exceed the amperage allowed by the circuit breaker.