Electrical – Two white, two black, two ground, and red wire in ceiling box connected to switch

I'm trying to finish installing a receptacle in my ceiling started by an electrician. All the wiring is ready, but I just need to connect the outlet. The receptacle is between two can lights in the ceiling, the furthest from the switch doesn't currently work, but I think I know why. Here's what the setup is:

In the one box, there are two sets of cables coming in. From the switch (left) side, there's one white wire (LW), black wire (LB), ground wire (LG), and one red wire (LR). From the other side, there's a black wire (RB), white wire (RW), and ground wire(RG). The light to the switch side will switch on correctly, but the light on the side where the other wires are coming from will not. I also want the socket between the two to be on all the time (or at least one outlet to be working all the time). The wires are connected with pigtails as follows:

LR + RB

LW + RW

LG + RG

LB

And from the ground wire there is another exposed ground wire coming from the pigtail, and there is another short white wire that I believe was connected to the white pigtails, but fell out when I grabbed the wires initially. The short white wire hasn't been stripped on the end, and neither has the black wire on the switch side.

I assume the correct configuration is to strip the left black and pigtailed white wire and plug those into the receptacle, along with the pigtailed ground wire, and then plug whatever I need to into the one active outlet. I assume the red wire is connected to the right black wire in order to pass it power when the switch is flipped, but not to control the receptacle, as it should be on all the time. From checking the wires, the left black wire (the one that hasn't been stripped) is hot all the time when the breaker is on, but the other black wire is not (I assume because it needs the red wire to give it power).

So I'm wondering if I'm allowed to pigtail those three white wires together, and then plug that into the receptacle, and if my other assumptions are correct before going forward. I don't have any experience with this, so I'd like to get a second opinion before I start plugging stuff in.