Ordering my hardwood this afternoon (~500 sq ft of Mullican RidgeCrest 1/2" engineered, going to staple down) and I need to figure out how much trim/threshold I need to order.

1.) Trying to figure out the best approach for the metal door threshold at the front entry door (flooring perpendicular to the threshold). The threshold is the square type (i.e. the inside-facing side of the threshold is vertical, perpendicular to the floor surface) and I'd say there'll be at least 1/4" rise from the hardwood surface up to the threshold once I get it installed. A couple searches have yielded various suggestions from butting the flooring to within 1/8" and filling with color-matched caulk, to using variable threshold with mitered ends and returns to the door, to undercutting the threshold and slipping the flooring underneath, to replacing the threshold altogether with something more "hardwood friendly". Unfortunately I don't think replacing is an option because I can't figure out how to get to the screws on the threshold to remove it, and I'm having a hard time picturing the mitered-with-return solution in my head (anybody got a picture?), so I'm honestly leaning towards just butting the flooring up with 1/8" gap and filling with caulk.

2.) Similar to #1, but with the door to the garage in the middle of a wall and flooring running parallel with the metal threshold instead of perpendicular.

3.) How do I finish the edge where the hardwood meets the carpeting on the front of the first riser on the stairs to the second floor? Can I just run the flooring right up to the carpet, leave a little gap and let the carpet pile hide it, or do I need to put a piece of quarter round along that edge between floor and riser?

4.) Flush reducers or variable threshold for transition into bathroom and laundry room which are vinyl? I know flush reducers are generally recommended, but I feel like maybe a square nose would work too and help to close up the gap between the bottom of the door and floor.

Sorry for all the questions lately... I'm planning to give back a little by documenting my install for others, once I actually get it done All the knowledge shared is much appreciated!

Personally, I would fit tight to both doorways, given that you are using an engineered product. For mental comfort, you could leave a bit of extra space at the opposite end of the floor runs in those two locations, but as long as you maintain proper environmental conditions, I dont think this is necessary.
At the stair, you could just fit the boards tight enough to compress the carpet a bit, and this will still leave you some expansion tolerance.
A reducer with a lip along one edge might work at the vinyl locations, or a t-mould. Or you could undercut a piece of the flooring to angle it down to the height of the vinyl, and use that as your threshold.

Alrighty; butt it up to the threshold and caulk it is!

One more question: I've got a small area of tile on the floor right in front of the fireplace, and after pulling back the existing carpet and taking a look I see that it's 5/8" thick tile sitting directly on the subfloor. I plan on leaving it there, so I'll need a transition piece around the edge between the tile and hardwood. Problem is I'm going to lay 3/8" underlayment on top of the subfloor, then put down 1/2" engineered flooring, which brings the finished height of the hardwood to 7/8" above the subfloor, i.e. 1/4" higher than the tile. I was originally thinking T-moulding, but since the heights are different the T-moulding won't sit flush (probably tilt in towards the tile leaving a gap underneath on the hardwood side). I've heard of offset-height T-mouldings, but Mullican doesn't offer them and I don't know how I'd match the color if I got it from another manufacturer. Is my best bet just to go with a variable-height threshold, and perhaps rip a little off the bottom of the deeper side since the height difference is only going to be 1/4"?