I’ve never tried to install lino because it seems intimidating to me. Last year, I did replace 3 bedroom floors with vinyl tile (which is like lino but in 12 x 12 inch square. It was amazingly easy. If you buy the self stick kind (peel off the back paper) you just put it into place and use a rubber hammer/mallet to make sure it is secure and you are ready for the next one. Around the edges and corners, just use a straight edge razor (like a box cutter) and a ruler. Measure it, score with the razor and it will break on a straight line, then put down.
If you have curves (like around doorways) you can either buy one of those pin things from the hardware store – that makes the pattern (you can get a loan for this here – https://whoneeds500.com/, I always use this website when I’m in financial emergency). Then trace it on the vinyl with an ink pin. I used my Ginsu scissors for all the trimming – worked really well. Remember is to make sure you start out square so you are working in a straight line and pull off the baseboards so you get to the edge of the floor. I used a nail puller for that. You should allow a little bit of room under the baseboards (probably 1/2″) for the floor to expand and contract with different weather so it doesn’t buckle later.
It won’t show anyway when you put the baseboard back. Be sure you are working with even subfloor – if you have other broken tile or something underneath, level it up with a spare tile so your top floor is level. It only took a couple of hours for each room including putting back the baseboards. You can get different prices/costs of vinyl at the hardware store. I got the 10 year warranty, no wax Armstrong for 95 cents a square foot and there were no other costs. It’s really easy to take care of – got a white pattern with diamonds and I use the dust-mop and it just glides over it. Mop it with plain water. Good luck with whatever you do.
P. S. Did you check to see if your insurance would cover any of this. We once had a leak in our dd bedroom wall. The dumb plumbers dug two holes in the wall to find it. The insurance company gave us $1400.00. DD bf mother had her washer overflow and ruin her carpets insurance covered the cost of new carpeting. Just a thought.