Flooring – How to remove a stair tread under linoleum without damaging the linoleum

Instead of trying to work with the flooring in a cramped space, I use a roll of craft paper and sticky tape to make a template of the room. Start in the center of the room, lightly stick a strip to the ground, and cut it at the edges of the room. You get a cleaner edge by going a little past the edge, then folding the cut edge back and sticking it down with the tape. Keep sticking more strips to the template until you cover the entire room.

While you're doing that, lay the flooring out somewhere that you do have the space to work. When it's ready, put the template on top of it, and cut around the edges. Regular scissors work well for vinyl flooring, and I know I've installed real linoleum, but can't recall if scissors were OK or if I had to use something more heavy duty like tin snips. Now that it's cut to size, you can easily carry it to where you need it and unroll it.

I don't see any nail-heads or screw-heads in the carpet edge trim, so you probably have one like this:

You might be able to save it, but they're cheap so I'd just replace it with a new one. Pry it up taking care not to damage the door trim. The carpet in the door opening should be loose enough that you can get access to the linoleum underneath (there may be a tack strip across the opening but you should still have a little room to play with). Finish removing the old linoleum flooring, then install the new flooring, gluing it down as you would normally. Remove the old edge trim from the carpet -- I'm familiar with ones that are folded onto the carpet edge to hold it, so just bend it back. Install the new one at the edge of the linoleum. Push the carpet into the new edge trim, making sure it's taut, then bend the trim down to hold the carpet; use a piece of 2x4 cut to the width of the opening so that it bends evenly.