I have run into the same situation many times. Let me say first, that I don't condone doing work that requires permits without them, as it often leads to crappy quality or safety issues. I always prefer to do quality work, but occasionally budgets make it necessary to do a quicky fix.
In your case, the most important thing is to strip away all old wax and dirt on the existing tiles. Tack down any loose tiles, especially around the edges with ring nails or a power stapler.
There are many de-waxing products and can be found at any box or hardware store. An alternative cleaner is ammonia and water. Cheap and effective, but smelly. You can also use a mix of regular bleach, TSP, and water. (BTW, never mix bleach and ammonia!!!!) Clean the existing tile with one of these products and scrub the surface well with a mesh type pad, like a 3M green scrubbie. Be sure to rinse the area well with clean water and allow to dry.
You may also consider a tile primer. This product is a bit pricey at over $30 a gal, but when applied over most any hard surface gives an excellent base for any self adhesive tile product.
When you get to the bathroom, remove the toilet and cut your tile around the closet flange, then reinstall the toilet. It is only two bolts, one water connection, and a replacement wax ring. It will look 100% better and avoid tiles coming up due to the toilet sweating and wetting the edges of tiles cut around the base. Only takes a few extra minutes, but will make a big difference.
If these are minor holes that don't cause loss of function of the floor, I think they are probably "wear and tear" that is a legally mandated part of every rental agreement (in the USA), so you could not worry about it and leave them be...
Per your question, properly glued this is way the professionals patch. I have noticed that they tend to put a little fillet of glue around the border if the patch, I suppose so you are less likely to catch the edge and rip it back up. I'm not sure what the glue is - sorry.
Cut the patch piece first (in rectangular shape) then lay it over the bad spot and use it as a guide to cut the damaged flooring. This will help you get a patch that's the exact size for the hole.
Best Answer
There is no way to fix that. Just replace the bathroom flooring. A small area won't be expensive if you DIY.
Maybe wait until you are ready to move if the dog could repeat this. It wouldn't hurt to talk to the landlord. If the flooring needs to be replaced, they might prefer something different, in which case they might agree to just taking an amount you would have spent on the repair (without you having to be responsible for the work).