After 40 years in re-modelling and building, this is one of the most common questions.
1-pull the peel and stick off,use a heat-gun if you have to.
2-If you see a solid piece of "sheet goods" underneath you are lucky. This should also be removed, but do NOT use a heat gun. The fumes are toxic.
3-There may be a 3rd floor-but I doubt it,
4-If you see your wood sub-floor you are in-luck!
5-Laminate floors look nice and they actually DO take water just fine, but make sure they are installed by a pro. The downside is that one bad scratch or dent is almost impossible to hide without removing a large part of the entire floor and re-installing it.
6-Yes, tile takes work, but I have found people love the look if done right. Cold? Well maybe, but in the summer I bet your pets will be sleeping on it,
7-The most durable surface and easy to apply that will last 50 years is VCT tile. Just like you see in most commercial hallways. But most people don't want that appearance at home.
8-My personal choice is Hardwood.
9-you can skip step 2 if you use hardwood.