It seems to me that trained monkeys answer questions when they don't know what the @3%$ they're talking about.
First, while linoleum is still available, it is very hard to find and is rarely used anymore. What IS used is VINYL flooring, either sheet (roll) or tiles.
Second, only a monkey would suggest that you leave any tiles that are too difficult to remove (the poor baby). Either you take them all up, or you leave them all down!
Third, self-leveling products compensate for minor imperfections, not a floor that has dropped due to settling. To remedy that, you either need to fix the subfloor properly, as someone recommended, or, if it is a CONCRETE subfloor (cement is one component of concrete!), float it out, which is not something a D.I.Y.er should try.
O.K., now that that's out of my system, here's what you need to do.
First, determine what your subfloor is by removing a couple of tiles. If you see wood, drill a hole through it to make sure it is an actual wood subfloor, not a wood substrate laid on concrete.
Next, you should repair the floor as described above, unless you can live with the slope.
Then, if it is concrete, you can lay your tile directly on it AFTER you remove the old tiles and any adhesive, and the wood substrate if there is one. (Fill any small dents, holes, cracks, with thinset first, and let dry overnight).
If it is a wood subfloor, you have to lay down an underlayment such as Wonderboard to make the floor more ridgid, otherwise the tiles will crack and/or pop as the floor moves when you walk on it.
There are lots of websites with detailed instructions (and photos) that can help you.
GOOD LUCK :)