First, I don’t use the backstabs, the quick wires holes you mention that you push wires in to. My personal belief is that they should be outlawed because of all the failures I have seen.
You ask if you can use both the screw terminals and the back stabs. Yes, it is code compliant with 14 awg wire. 12 awg should not fit in the backstabs. Backstabs are only rated for 14 awg wire, which should have a 15A breaker.
A better way to make the connections would be to add a 4th wire if 14awg and a wire nut. The extra wire then goes to the receptacle, this is called pigtailing and is a superior method to using the backstabs. Then only 1 wire to the receptacle. This the method I have used for my career.
Another method is using a device called a lever lock, they are newer and I believe for DIY they are even superior to wire nuts. Lever locks are rated for both 14 and 12 awg wire.
Lever locks cost more but are really easy to use:
- strip the wire
- open the lever
- slip the wire in and close the lever and it is locked.
You need one for each wire, and they come in 2-6 hole versions. They take no real skill to use and for that reason I think they are a better choice for most DIYers. I started using them on lighting fixtures about 6-7 years ago.