The first thing would be to research the building codes for your jurisdiction. This will dictate what you can and cannot do with regard to this project, including applying for the appropriate building permits. I am adding an addition where I currently have a carport (my existing roof extends over this space). In my case, we will have to dig down under the existing slab to determine how the existing roof is being supported. What we find, along with the building codes, could change the scope of the project. And just because it exists, does not mean the previous owner did the construction correctly or current code will apply. Good luck and expect the unexpected....

I'm not sure who is advising this but it would seem odd. As long as you don't have any visible water problems now, then a vapor barrier over the existing concrete (I just use 4mil plastic) will be fine. You may wish to put down wood sleepers (always PT even with the plastic) and some rigid insulation between below the subfloor.

Your toilet requires a 3" drain which is probably the size they would run over to your house to tie into the existing. Both the sink and toilet would also be tied into this same drain under the slab. With regards to your concerns about the slab cracking from saw cutting. No, this shouldn't be an issue. As it relates to your slab and the foundation, you are correct, this we could not tell without seeing pictures or having a better understanding of what your outbuilding is. In the grand scheme of things as it pertains to the plumbing specifically, the only thing the foundation will really play a roll in is the plumber will either have to drill through the wall itself to get out of the garage, or dig under the footing itself if there is one. The second scenario may not be feasible as you most likely would have issue getting grade over to the existing drain you will be tying into. One thing to keep in mind is the number of existing toilets you have in the house. Depending on the code your local area has adopted, if you only have a 3" drain going out to your sewer/septic, you can only have 2 toilets on that drain (some jurisdictions will give you a pass and allow a 3rd. If you already have 3 toilets inside the house, make sure your drain out to the street or septic is a minimum of 4" or you will be dealing with up sizing the main drain as well from your house.

that's a huge deal. if you just paid to have this done, they will have to fix it. you never want sitting water at the base of the structure. ever. the way is is now, you'll have water seepage into the basement, mold, and rot. nope. have it redone. I can already see in the picture how it flattens out from the drain to you house. there should still be slope all the way to the wall. get a level and span it from the house to the drain. if the level is flat for the first foot or so, you have a big problem. I believe the slope is 1/4"for every foot, starting from the house and moving outward. So if your patio is 20', you should have 5" of slope. (however, they put the drains in less than 2' from the house. so the slope is going to differ in order to funnel the water to those drains. don't know why they did them so close. if they give you grief over it, call in your own inspector to give their opinion.