Choosing a starting wall so that the the detailed work on the finishing wall is hidden in a closet; another way to start laying out your laminate floor; professional installers demonstrate a second way to begin a laminate installation.
With a door in there it looks like you would only see a little bit of the threshold, which is perfectly normal. This is how I have it in my house where my tile bathroom meets the hardwood floor. The threshold is not "hidden" by the door 100%.
Use your hammer and prybar to remove the floor trim from the closet floor. Don't break it, so you can re-use it at the end. Spread your flooring felt over the floor, cutting it to size with your razor knife. Staple it down with your staple gun.
Today I started laying some hardwood flooring in one of my bedrooms and initially when I ran into my closet I thought, no problem I will create a threshold parallel to the floor joists (in the closet only, every where else I run perpendicular to the joists). ... Can I install a row in this closet door threshold where there is little to no room ...
Laminate flooring is made to move when needed, so screwing through to the subfloor under the laminate may not do the floor well. Since sliding doors do not exert a lot of pressure at the guides on the bottom, you could size the screw to only secure itself into the laminate itself, say no more than 1/2" long screws, considering the laminate should be about 3/8" thick, the mounting plate is about 1/16" thick and the pad under the flooring adds a little too.
Lay a piece of scrap flooring face down next to the closet door trim. Cut the door trim off flush with the top of the scrap floor piece using the oscillating saw. When the cut is finished, a piece of floor should slip under the door trim for a finished look.