Hello everyone! I’m preparing to glue down some 5-1/4” engineered on our concrete slab. As a complete newbie, I am curious what the correct way is to install the flooring in a transition-less fashion. Mainly, we have a hallway that
has 3 bedrooms off of it, and the hallway also opens at one end into an open plan living and dining room.

Since the hallway is very visible, I wanted to square up the install to this hallway (equivalent-ish width end pieces against both side walls of the hallway). What I’m confused about is how to then keep running the floor into the bedrooms, while also squaring up these rooms (i.e., avoiding small slivers or unequal width end pieces). Should I be ripping down the first few boards that enter into the room to compensate for the room being out of square and potentially not square with the hallway? If so, are there any tricks to make the different width boards less visually jarring? Or, is there a better way to do this? Also, as the hallway opens up to the living room at one of the ends, how do I continue the installation to square up to the living room?

You can see a diagram of our house here: https://imgur.com/a/3Qvxv0W

As a flooring contractor we generally start on the longest straight wall to help eliminate extra work. I do not try to center the pattern in a hall so side pieces are equal. Since the floor is going to be glued you can work away from the tongue side or the groove side with equal results of fastening the flooring down. Since you are working with a board width that would pass through a doorway which is usually 4 and one half inches because interior walls are going to be 2X4's this would make the starting layout more important so boards at doorways can be difficult and usually result in a rip along the wall of the next room.
I would start along the wall of the hallway next to the bedroom doors. Run a chalk line close to 5 and a half inches from the bedroom wall all through the hall and into the living/dining rooms. In larger rooms, out of square walls are not so noticeable with furniture in the rooms.
If you are not using pre-finished flooring it is possible to plane the groove edge a little and still have the tongue fit as it should so over a few rows. You could bring an out of square room closer to correct alignment without noticing a small taper over the width of these boards. Sometimes you can almost get an eight of an inch off the edge and still fit.

Thank you for your reply Pete A! It is very helpful! You mentioned trimming down the groove end. Would something similar be possible with prefinished boards?

Pre-finished flooring almost always has a"micro-bevel" on the edges of each plank. When this is disturbed it will show up in the reflection.
You may get away with this using a belt sander on the edge, but I think only near a wall.