This is related to what is known as separately derived systems and non separately derived systems, and it concerns avoiding unnecessary impedance on the grounding system that would adversely effect over current protection, i.e., breakers and fuses.
If the neutral is switched along with the phase conductors in the transfer switch, you have a separately derived system and you would NOT separate the grounds and neutrals in the generator only, which contains the main over current device "main breaker". You WOULD separate the grounds and neutrals in the sub-panel. Again, this is because the main over current device, "main breaker" is contained outside in the generator.
So, in your condition the generator is portable and thus does not fall under the permanently installed generator rules. But if it were, you would want to separate the grounds and neutrals at the generator and sub panel and be done with it.
To address your main question whether your current setup is safe. As mentioned above you ideally want the lowest impedance for ground fault protection. But by not separating the grounds and neutrals in the sub panel you are creating MORE impedance.
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
It went further to the center as long as it did not push the one to the right out of place don’t worry.
If you want to this is an easy fix but not needed in my opinion (I am a professional electrician).
There are screws that hold that face plate in position. Remove the lower screws first when you only have the top 2 left press in on that cover to hold it in place and remove the last 2 screws.
Pull the cover (called a dead face away) set it down. Turn that breaker off then pry the handle towards the left and out they just snap out. It may slide at this point prior to pulling out.
Look around at the wires if you see burned insulation let us know with another photo, if every thing looks good carefully put the panel cover back on I usually do the top 2 screws first opposite of removal put all the screws in make sure the breakers are all on some times you bump one and it will trip to the off position. Always move a tripped breaker fully to the off position and back to on. Most when bumped or turned off will go to off but a light pull to off then it should move easily to the on position.
Give it a try, don’t touch any of the wires going into the main or any breaker these are the hots, I am sure there will be a U tube video on replacing a breaker if you want to watch it but it is a simple job you can do if you want but it is fine.