What do you mean by "level" the floor?

Of course you can lay the new on the old. A solution for the molding is to put quarter round down. A little more work but not much. With regard to problems seeping into new tiles. This can happen. My vinyl has a slowly growing stain. In time that damage will make its way to the new product. Personally, not worth taking a chance. I would rather remove the old and put the new stuff down fresh. There are many shows where they remove several layers of peel and stick so it is obviously something that is done by many. Whether it is proper installation or not is another question. My dh has a saying we stick by, not trying to be rude or anything, but if you are going to do a job right, do it right the first time. Most of our jobs end up taking longer, costing a few dollars more (which I do not always have) but in the end come out beautifully.

Wow. Ok. Where to start. First and foremost the SHEET vinyl is the only one that can brag about water proof. If you have leaky pets or leaky people, this is the go-to product if water PROOF is a requirement. Sheet vinyl requires a professional install which is normally quite reasonable. The costs grow with the subfloor PREPARATION costs (where are a separate line item that may not have a "final total" until the job is complete). The click together vinyl planks are all the rage right now. Some are DIY friendly - but most are quite picky about technique and take +++patience to learn how to do it right without breaking the click-edge. It is possible to do this yourself....but it requires an understanding that you will have a STEEP learning curve. Be OK with that. The cost to install the planks is a little bit more than laminate flooring. Again the costs go much higher once subfloor preparation is added in. Again, this is a separate line item that, as above, might not have a final tally until the job is complete. The "flatness" quotient for vinyl planks is extremely important and can be considered outside the comfort zone of many long-term DIYers who have tackled several different projects. Again, it is possible to DIY....but it takes quite a bit of time to learn the skill/trade. A professional can have most of this done before you have figured out how to work the self-leveling-compound (which is not self-leveling BTW). Is there anything else that you were looking at for information? I've just touched on the two biggest differences between the two products. I didn't know how much info you were looking for.

I would also suggest avoiding vinyl at all costs - especially given the beautiful hardwood in the remainder of the house. Ceramic will always wear better, is better for resale value and a healthier product for your family to come into contact with. A charcoal grout update and charcoal walls or a fun graphic wallcovering would be nice budget friendly updates to this space and still in keeping with the quality of the other finishes of the house. Best of luck!