I saw a central R/A installed in a new home without vent boot/register box on the opening ceiling, and the R/A duct installed OFF SIDE from R/A opening ceiling (please take a look on these photos on this website www.thebuilderfromhell.com/photos for more details). I wonder if this R/A is properly installed? This look like a "wood frame" (as picture shows 4 sides above a R/A opening ceiling) is using in lieu of register boot. Should this off-side "wood frame" to be sealed/installed with R/A Register Boot? Please advise!

Looks like a PPP job. That's P#** Poor Planning. It's not the prettiest thing I've ever seen, but not particularly uncommon either. Especially if there is a space limitation that prevents installing a regular formed boot and it looks like that is the case here. I'd sooner have a wooden box than a bunch of sawed off unheadered joists. At least they sealed the wooden box. It is a return, so it's not like condensation is going to be an issue.

I assume that this is located within an attic area? I hope they put enough insulation around the outside of this box. Your contractor will be out of business in a few years at most. The service complaints they will be getting from what I would think is their typical type of workmenship will kill them.
The bigger issue here is that the local code offical allows this poor quality workmanship. But his or her hands are tied. Cold air returns can be pretty much any material and because this fellow does not have a metal fab shop he cut corners by using plywood and lumber.

Many years ago I operated a HVAC company. It was folks like that who underbid me so much I could not make a fair living. Now I just inspect them and tear them a new one when I see this type of work going on.
If this is just the return, I cannot wait to hear back from you about the air balance issues and no airflow your going to have within the house.
Get another contractor in and fire this guy if you can.