The type of cable you linked to is SE cable covered by Article 338 of the National Electrical Code. It connot be used underground with or without a raceway. Here is the pertinent code language.
338.12 Uses Not Permitted.
(A) Service-Entrance Cable. Service-entrance cable (SE) shall not be used under the following conditions or in the following locations:
(1) Where subject to physical damage unless protected in accordance with 230.50(B)
(2) Underground with or without a raceway
Notice item 2.
You could use several other types of insulation in a raceway, including THHN, XHHW, XHHW-2. USE is for direct burial and can be used without a raceway except where exposed to damage so you need to sleeve it in PVC or metal raceway above ground.
Aluminum also needs an anti-oxidation compound such as NOALOX or ALNOX, there are many brands.
Rigid or IMC conduit can be buried only 6" deep. Table 300.5
SE cable can be used as a feeder for interior wiring so if you kept the feeder inside the house and garage without going outside you could just protect it with PVC in the garage and be good. Or you could run the cable outside just not underground.
You will need a single 100 amp 2 pole breaker to feed the sub panel. 2 -50 amp breakers is not acceptable by the Code.
A separate disconnect is not required. The panel in the garage is not a motor so the "within-sight rule does not apply". The breaker in the main panel can act as a disconnect. You actually don't even need a main breaker in the panel if you don't want one. You can buy a Main Lug Only panel if you wish. The main only panel you linked will do just fine without a main.
Good luck and stay safe!
Fat aluminum and fat conduit are your friends here.
We'll start with the conduit -- Schedule 80 PVC won't go anywhere when you put it in the ground, is plenty rugged enough to survive what a homeowner can toss at it when brought aboveground, and is available at the orange borg, in sizes up to 3". Since the electrician won't be there to bail you out with the truck o' pulling tools anytime soon, go with the 3" size even though we're only at about 10% fill here. Bury the conduit to 24" -- that will guarantee your 18" of topcover. As to the terminations -- it will come up the outside of the buildings in question into LBs that then go into the building sides.
Then, we pull out the jumbo aluminum wire. For the price of your proposed 6AWG THHN/THWN-2 copper, you can get 1/0-AWG(!!!) RHH/RHW-2 AA-8000 series aluminum, and since this is a feeder, it's going from aluminum breaker/panel lug to aluminum breaker/panel lug anyhow -- if your insurance complains, tell them to go re-read NEC 310.106(B). That aluminum, by the way, will carry 120(!)A -- 100A is a more practical size, though, so we'll use that instead. With that, you can use an 8AWG bare copper ground wire.
Sorting out your subpanel
You'll need a NL20 lug to land the ground wire on (instead of that kludge-screw) and a BR2100 breaker with a BREQS125 hold-down kit to serve as a submain. The orange borg can order all these in if they aren't already in stock there. You'll also need a torque screwdriver for this.
The hot wires, then, will land on the terminals of the BR2100, which is your service disconnect (you can backfeed any breaker that does not have LINE/LOAD markings on it). The neutral lands on one of the big neutral bar lugs, and the existing kludge-screw is replaced with the NL20, which is where the ground wire is landed. Make sure to torque all connections to spec and use anti-oxidant on all aluminum wire terminations so they don't come loose and become a sparky mess later!
Landing the feeder in the main panel
If you want to bring the new feeder up to its full 100A capacity, you can put another BR2100 breaker and NL20 neutral lug in the main panel in place of the BR240 that is currently protecting your feeder, terminating the ground on the existing bars. Of course, if you have a big lug spare on your main's existing neutral bars, feel free to use it, and you can run the 1/0 at 40A by pigtailing it with Al7Cu or Al9Cu lugs to smaller wire if you wish to postpone the main panel rework until later. Again, remember to torque all terminals to spec and use anti-oxidant with the aluminum wire!
Getting grounded
You'll need a ground electrode at the shop since it's a detached structure. Contact your local electrical inspectors (aka Authority Having Jurisdiction or AHJ for short) and they'll be able to guide you on what to do given your atrocious-sounding soil conditions. (For instance, a concrete-encased or "Ufer" ground would work if you had reinforcements handy to ground to, but not every pour has such.) Simply shoving copper rods 8' into the ground may be a waste of copper, though, in that it may not yield acceptable results in your situation.
No matter what you put in for a ground electrode, it can be connected via more of that 8AWG copper to the shop subpanel provided you protect it from mechanical damage. The neutral and ground being kept separate at the subpanel, though, is absolutely correct despite your seeming bewilderment at this discovery -- neutral is grounded at the service entrance and nowhere else.
Best Answer
Yes, you should immediately downbreaker to a 135A breaker. Hold on. Those aren't made.
Oh dear. Well, per NEC 240.4(B)(2), you should round up to the next available breaker size.
You'll need to fit a 150A breaker ASAP. Sorry to put you to the expense! :)
Seriously, there's one other thing that may apply here. Since all the power for the entire domicile comes through this feeder cable, it actually gets to be derated like a service. Per 310.15(B)(7), you apply a (favorable) 83% derate, meaning the 150A breaker only needs wire rated for 124.5 amps.
Alternately, take your 135A wire ampacity and divide by 0.83 to give 162.7A if it's handling all the power for a domicile. That again rounds up to 165, 170 or 175A depending on which breaker sizes are actually made.