LVT is great. Not soft at all. Been fitting it nearly 40 years. Have it in my house. Looks real if you go with the right one.
Large tiles look at , floorify & kahrs.
Yes then another coat on top.
Best to use the same as underneath but you can use a different one if it’s a similar compression strength
So a system we use a lot
Ardex na / Bostik C955
F77 Dpm
Ardex P4 primer
Ardex CL
That looks like asphalt to me which is good.
Need to get it clean 1st. Best use de greasing crystals from Screwfix.
Then fit in holes with ardex na / fball 1200 pro / Bostik c955 advanced /
Then a liquid Dpm over them sections and any cracks
It’s sloppy and loads of fitters do it that way as it’s quicker. But it leads to heel & toe wear. It’s probably not grippers correctly and that’s why it’s loose as well.
Also you need a clean sub floor if using a water mix smoothing compound. No contamination of old adhesive , paint etc. Also if older then 1980 you will probably need a liquid Dpm.
Have a look on the all in all trim website for cheaper trims. Are all the doorways short of the doorway like the picture ? You will need to notch the trim around the door moulding.