I am installing about 1000 sq ft of flooring on the main floor of my house. I have a few general install questions, it's been a long time since I've done hardwood flooring. (15 years and it wasn't a big project).

Most of the end of the flooring will go underneath baseboard... not much of a problem. There is a metal transition on one end where it meets a tile floor. (see picture)

There is a part (see picture with two yellow arrows) where it will start in a stair nose piece, and end butted up against a tile transition. I was trying to keep the sleek look by using the tile transition pictured (see blue arrow). After install I realized this may make the hardwood flooring a bit trickier because I won't be able to "hide" the end under anything.

This must be a fairly common setup, but I can't find any good info on it. Currently my hardwood that is coming out has a perpendicular piece running at the transition with the blue arrow. What is this called? Also, I have to cut the pieces on this end, so they won't have a grove (right?) ... so in this setup, those pieces aren't inserted in to the perpendicular piece?

Also, I am going to switch one room to carpet. Do I use a perpendicular piece to end the hardwood? Can I transition straight from that piece to carpet? I like that look as opposed to using a transition piece.

This is 3/4 inch hardwood, 5 inch wide.

Thank you

Pictures:

https://ibb.co/album/7Ycj87

I call the perpendicular board across a doorway a header board. If the ends of flooring can not be fitted into a groove or is missing the tongue you may help the piece stay fastened with some sub-floor or construction adhesive, besides the fastener type you use for the rest fo the floor.
the tongue that fits into the groove on the ends of boards is not milled as perfectly as on the sides of the flooring so there can be a lot of play. If you would like to have more stable joints where the ends meet you can staple, cleat or nail the ends of the boards where the tongue is, then slip the groove end flush..
The 3/4 inch thick flooring can transition directly to carpet. The carpet is either turned and tacked or installed with a tackless strip and the edge is compressed into the 3/8th inch wide gap between the tack strip and the hardwood with out staples.