Given the location of your water supply, I think you did well to find a toilet that fit in that spot, so let's help you make it work.
With regard to your questions that haven't yet been answered directly:
The 3.5" off the wall is not by itself a very big deal. It's a function of the fact that you put it on a 14" rough-in (so there is instantly 2" more than "planned" behind the tank). The same thing that helps make the thing fit in front of your water supply is necessarily giving you a bigger gap between the tank and the wall. A Toto Drake usually has close to 1.5" behind the tank when installed on a 12" rough, so when you add the 2" from the bigger rough-in, you're at about the same place. The toilet in my rented apartment has the same issue of a big gap behind the tank, and we have long since stopped noticing it.
There are few toilets that put the tank against the wall and none that I know that require the use of the wall to stabilize the tank. (There were/are wall-mount tanks that are secured to the wall and coupled to the bowl with a pipe, but that's a different product.) One reason that close-coupled toilets are not designed to stabilize against the wall is that on most brands there can be 1/2" or more of variation in the size of the final china product when it comes out of the kiln. Design the toilet to have the tank right up against the wall for stability, and you are asking to throw away a lot of toilets that won't fit in the space, or, conversely, won't touch the wall. Similarly, not all 12" rough-ins are going to be exactly 12" from the finished wall; it's not just sloppy construction; it can also be that someone tiles the wall after the flange was installed 30 years ago. So, again, it doesn't make much sense to build a toilet for mass production that is going to need exactly 12".
Your tank is unlikely to be designed to wobble, at least to the point that you are probably finding it to wobble. That thick rubber sponge gasket on the bottom of the flush valve nut is designed to be compressed. As long as the china isn't touching china, you stabilize the tank by compressing that gasket straight down.
I don't have the instructions for your toilet, so I don't know where the china pieces are supposed to touch or come close to touching. I would follow the directions. Particularly, I would use one hand to hold the tank totally-vertical and tighten little by little. If you can still slide a couple of pieces of paper all the way around, you're not going to crack it -- but again, I don't know this toilet.
One way to make it more likely to compress the sponge gasket properly is to use a double-nut setup. That is, get a set of tank-to-bowl hardware that has -- for each bolt -- at least one rubber washer, two metal washers, and two nuts. You put the rubber washer under the bolt head inside the tank (and ONLY the rubber washer, not a metal one), then metal washer and nut on the outside and tighten finger-tight plus 1/2 turn with the wrench. The bolts will stick out nice and straight (i.e. perpendicular to the bottom of the tank), and you'll have a good seal inside the tank before you even mount it. Then you place the tank on the bowl, put the second metal washer and second nut under the bottom of the bowl, and use those to pull the whole thing straight down, little by little, alternating sides. This is how Toto tanks are mounted. It makes it easier to get it straight, among other things. If you get a second rubber washer, use it under the metal washer on the bottom of the bowl (i.e. rubber washer, metal washer, nut, in that order, under the bowl). Or toss it; it's not really necessary. I like the sets that use two identical regular nuts on each bolt, rather than the ones that have one regular nut and one wing nut. I think the regular nuts are easier to tighten using a ratchet or wrench than are the wing nuts. (If you have the wing nut, it goes under the bowl not under the tank.)
Hope this helps. You have to consult your instructions, but I found in my first few installations that if I am super-careful about making sure that china isn't touching china, and the tank is straight, I can tighten more than I initially expected on the tank, such that it's sitting nice and firm on the sponge gasket and not moving around.
So we are clear, on most designs it isn't china touching china that is keeping the tank wobble-free. It is the compression of the tank firmly against the sponge gasket and the support that that gasket gives to the center of the assembly. (Certainly, most manufacturers tell you to tighten until china just makes contact with china, but the fact is that the tank will seem pretty darn stable even when you can slide something the thickness of a business card or a couple of sheets of paper all the way around the installation.) As an example, I have a toilet that's more than 50 years old that I rehabilitated with new working parts. When I went to put the tank back on the bowl with a new sponge gasket and hardware, I realized that the plumber that had installed it years before had stabilized the tank by using a bunch of plumber's putty under it. I scraped all that off and made a couple of tries to get the big, heavy old tank tightened down perfectly-straight without cracking it by overtightening. My final result is just as stable as the old tank was, even though it's just barely-touching the base, if at all.
Given this, you will get a little bit of play if you push or pull on the tank, but it will feel nice and solid, not wobbly. That's good enough.
