Linoleum
Linoleum
Unless you have several layers of old linoleum, most installers would put down a layer of luan to prepare the surface for the new vinyl. This will give you a far better result than trying to tear out an old floor. You may find a worse initial surface to deal with than going the luan route. If you have never laid vinyl sheet goods, I don't recommend you try. It requires several specialty tools, knowledge of proper trimming, and experience in laying out the pattern.
You can lay luan sheeting down to make a smooth surface and then install vinyl over the top. It may require repositioning any quarter round trim that may be down and cutting the bottoms of door casing. But this is usually the recommended way. The alternative is to pry all of the old flooring up and installing new subfloor prior to laying the vinyl.
Unfortunately there are no magical solutions for this other than a lot of hard work with a floor scraper (shaped similar to a hoe). Make certain that the old lino does not contain asbestos as it is hazardous. Once you get the lino up, you may be left with some glue residue which can be removed with a lacquer thinner or acetone. You absolutely must turn off all pilot lights and work in a well ventilated area.
You could try this: Clean the area as much as possible - underneath the slit as well, since it is likely that some dirt has gotten in there. Heat the area up with a hair dryer to give the lino some flex. Using a dull knife, push the slit down and together to see if it will fit tightly back in place. Providing it does fit, heat the area again, open the slit back up and apply some quick dry glue. Using the dull knife, push the slit back together. Apply some masking tape to hold it while the glue sets up. Once the glue is dry, remove the tape and seal the seam (follow the directions on the seam sealer kit).
Today's lino floors have come a long way in terms of pattern, color and finish options. It is possible to update to a more trendy look with either option (tile or lino). Cost of lino could start at a lower budget price - traditionally the prep required for a lino-to-lino installation would be less (although this is very case specific and would require an estimator to physically see the space to ascertain). However, some higher end lino floors can be very equivalent in price to a tile floor. Something to think about is resale - hard surface flooring (i.e. - tile, hardwood, etc) will add more to the value of your home than soft surfaces (i.e. - lino, carpet, etc). When renovating certain areas of the home, I would say that you should be concerned with working with the existing colors - not necessarily "matching", but coordinating with these colors.
Sorry to be the bearer of bad news, but you can't lift the lino from the subfloor without damaging the lino. Do you have any remnants of the lino available? If so, you would be able to cut out the damaged area, install new subfloor and a lino patch.
If it is glued down it is very hard to reuse it. Most likely it will chip and year when you pull it up.
No. Linoleum requires a firm substrate (i.e. - concrete or K3).
You can try a latex contact cement (low odor) however the lino has usually seen better days if it is curling badly and you may want to consider replacing it.
Any over the counter, quick dry glue can work - providing the area in question is small (i.e. - under 12 inches). Please note: this fix would be classified as temporary and will not be as clean looking of a seam as the original. For larger seams, I would have a lino repair person address the problem. The following sequence can help if you are attempting to do this repair yourself: Clean the area as much as possible - underneath the slit as well, since it is likely that some dirt has gotten in there. Heat the area up with a hair dryer to give the lino some flex. Using a dull knife, push the slit down and together to see if it will fit tightly back in place. Providing it does fit, heat the area again, open the slit back up and apply some quick dry glue. Using the dull knife, push the slit back together. Apply some masking tape to hold it while the glue sets up. Once the glue is dry, remove the tape and seal the seam (follow the directions on the seam sealer kit).

