You should never chain two GFCI circuits together. When you install a GFCI circuit, you should only ever chain standard outlets off of them. Multiple GFCI on the same circuit can cause each other to trip.
So yes, what you want to do is actually what you should do.
What you do need to do, however, is make sure you connect the new outlet to the LOAD terminals on the GFCI outlet. That will put the new outlet on GFCI protection.
You should also sticker the new outlets as GFCI protected.
See in the above how the duplex (standard) outlet is fed from the GFCI, NOT the main line.
And in case it's unclear- the 3 gfci in the pic are not chained - they are connected in parallel. Only the standard outlet is "chained" - or connected in serial.
If you don't plan on using any 20 ampere cord-and-plug devices, there is no reason to install 20 ampere rated receptacles. Circuits rated for 20 amperes are useful in the home, because they can support more devices. 20 ampere receptacles are often not advantageous, except possibly in a workshop or garage.
Best Answer
There are no limits on the number of receptacles per 15A or 20A circuit in general, though there may be in some jurisdictions. See, for example Is there an average number of outlets that are wired off of one circuit breaker?
A GFCI, whether as part of a breaker or combined with a pair of receptacles, is only monitoring for a difference between hot & neutral - i.e., watching for some current to go missing. As long as the total is less than the capacity of the GFCI device, which itself should be matched to the capacity of the overcurrent protective device (fuse or circuit breaker), it really doesn't matter how many receptacles or devices are connected.
There is a theoretical issue as follows:
A typical GFCI trips in the range of 4-6ma. You could have a bunch of devices that each leak 1-2 ma, which is generally safe. The cumulative effect of these devices on one circuit (which could easily be with as few as 4 receptacles, so you don't need "lots of receptacles" for this to happen, but the more you have, the more likely it will happen) would be enough to trip the GFCI. Yet when you go back to the "unplug everything and plug in one at a time to see where the problem is", the problem device might never be identified - and yet the tripping would continue.
However, I suspect that only a very small percentage of devices would ever have this low level of current leakage without gradually progressing enough to be a problem. Plus this could happen even with just a few receptacles.