Integrated sinks are great, but you have to like the materials they work with. They make soapstone sinks, but they are usually installed with a bit of a lip. If that overhang getting grungy worries you just take a toothbrush to it from time to time.
We were specifically told NOT to use vinegar on our granite…and by more than one person.
@jshain recommended Method Daily Granite Ckeaner for the daily wipings and it is terrific!
Re:sinks- so happy with my deep and roomy Franke “orca” shaped sink. Plenty of space for very large items. Hated having a double bowl with neither side big enough for a cookie sheet or large sauté pan. YMMV.
If an undermount sink has a negative reveal, there will be issues with gunk collecting under the countertop. A fabricator can do a flush mount or a slight positive reveal to address that issue.
In an integrated sink, as stated above, it is all one piece so there’s nowhere for crud to accumulate. It can be in stainless steel or other materials. I have never had an under mount sink, so I can’t speak much about them.
I just am very pleased with our stainless sink. It’s very easy care and can withstand temp extremes easily. It doesn’t stain either and is easy to clean and always looks great.
I have integrated sinks in my bathrooms. In kitchen, under mount one bowl SS sink, with positive reveal. I’ve never had a problem with gunk.
I do like farmhouse sinks, but they don’t fit my FL house.
We have a Shaws Original farmhouse sink in our kitchen that I love. One big undivided sink-- easy to move dishe, pots, and pans about. They are basically designed to have a half inch positive reveal and very easy to keep the “gunk” out. The one annoying issue with ours is the bottom always has debris of some sort. The almond color shows it off really well!
We have undermount bathroom sinks in our MBR with a negative reveal. As bookworm states, a royal PITA!
It depends on the granite. The vast majority of granites (true granites) are impervious to vinegar and other acids. Some granites however will etch. It’s best to get a sample from the stoneyard and conduct a test yourself to be sure.
Regarding sink depth and granite. The granite counter surrounding the stainless steel sink will add enough depth you may not want to go too deep. Of course you will want a quality metal gauge (18?, not 20, no need for the best) and 8" instead of 9" works well for me. You do not need a “name brand” or expensive sink- the middle of the line work well.
Actually, for SS, the lower the gauge the better. I suggest 16 gauge (not 18) or 14 gauge. The lower gauge is heavier and better resists scratching. Also, get a sink that is sound proofed on the underside to buffer noise.
My sinks are 16 gauge.
If you are going with SS, I like the Elkay sinks in 16 gauge.
We just had the kitchen redone. Quartz countertops and a single Blanco Silgranit sink. I love my sink. Can I say that again? I LOVE my sink!!! 
The great thing about Silgranit is that the color is all the way through so they don’t scratch or stain. You can’t scorch or mar them. And the large single bowl is wonderful. All the dirty dishes fit, including cookie sheets.
I selected one with the drain at the side rather than in the middle so that the dishes don’t block the drain. And I picked “Truffle” color, which looks amazingly good with the brown-toned granite counter.
Even my husband commented how much he loves the sink. He’d really resisted getting a single bowl rather than a double one and he freely admits he was wrong.
I picked quartz countertops because quartz is heat resistant, doesn’t need sealing, and is easily repaired. We picked Viatera Solar Canyon and get lots of compliments on it. It looks great with the off-white cupboards.
Another fan of granite counters and undermount SS single sink.
I recently replaced a large section of the kitchen counter and eliminated a second level that functioned as a serving counter. By eliminating the backsplash and overhang, I gained more front to back space for the sink.
The 19 year old SS sink was still completely fine but it had always been too small, so I went for the largest sink that would fit in the cabinet, gaining inches in every direction, including depth.
Well…what I did not consider, as it was never an issue with the last sink, is that the sink needs to be pitched, if that is the correct term. (The way my walk-in shower is pitched so that the water flows to the drain hole on the floor.) So while this new sink is lovely (see link below), food residue does not easily drain unless I move it by hand. That is a huge hassle as I used to be able to just spray water and residue went down the drain.
I never treated the granite countertops and they have remained in amazingly great shape, other than some pitting in the work area the the left of the sink. Since that section of the counters has now been replaced, I will treat going forward. Soap & water on a sponge keeps them looking clean.
Same with bathroom marble counters. There is probably something I was supposed to have done to them, but I haven’t and they are fine. Perhaps dark colors are more forgiving.
The granite counter in our hose was already at least 10 years old when we moved in, and I had no idea it was supposed to be sealed until I had been using it for at least 15 years. I still haven’t. It does not stain, despite hard use with all kinds of potentially staining things such as blueberry juice when making jam, turmeric, acidic liquids of all types (I used to make lots of pickles). I have been known to clean something stuck on it with a brillo pad. 
The only place it shows anything is immediately around the cooktop. I had to replace the original one, and the new one has a slightly smaller footprint. I think the slight darkening is probably due to 30 or so years of cooking grease penetrating under the metal.
I don’t know what kind of granite it is, but it is as hard as a rock! It is fairly light color, a very natural-looking speckled combo of shades of gray with black.
BTW, on the double vs single bowl sink, you CAN have a sink with two big bowls, if you are willing to sacrifice the counterspace.
Our kitchen granite is called verde ubatuba and I sealed it just after moving in to our 5-year-old home (at the time we moved in). That was 4 years ago. It didn’t need sealing then and doesn’t need sealing now. The granite in our last home got sealed every 1-2 years. It just depends on the granite type, some much more porous than others, which needs more attention.
@jshain – I am fairly sure my Uba Tuba was verde also, and it has been very resilient. I had to replace a long run of it and was dismayed to learn that today’s Uba Tuba looks nothing like the Uba Tuba of twenty years ago. Ended up selecting an entirely new stone for the new section of countertop and someone would need to point out the two different stones in place in the kitchen. They are not adjacent to each other but no more than four feet apart in one section.
I have been enjoying reading along this thread and just want to say it sounds like all of you have really lovely homes!
I have a huge porcelain farm sink and a small bit of butcher block counter on either side. Mostly I have antiques in my kitchen, and my “counters” are drop leaf/ gateleg tables pushed against the wall with leaves folded down till I need more work space.
I have granite counters and SS undermount sink. Mine is an Oliveri and I picked the one with the drain board because I hate dish racks. I love my sink.
@calla1 - I stopped at a stone shop the other day and as I was looking at the sinks, the secretary walked by and pointed to a sink and said, “I have that sink and it’s amazing!” I asked her what kind it was and she said it was a Blanco Silgranit ! She raved about it. It has definitely moved up high on my list and I loved the looks of it.