Your DH needs to text her that it isn't okay to just show up, if the kids need something that he will go buy it for them or she can coordinate with HIM a good time and location to drop it off, and if she shows up unannounced that he will call the police as he does not give her permission to be on your property. I would also send this via email so there is a paper trail.

Why didn't the stuff come with him when he came to stay for a month?

Re the Austin issue.  I can see how a kid would be interested if he thought his dad was moving to Austin.  Maybe or maybe not be spying for his mom as a move obviously would impact his life too.

Re the stuff drop off... she can leave it on the porch... you don't answer the door if you are home and certainly don't arrange to BE home.  The children are supervised so as to not open the door either.

Stop the drop bys! Text BM and say nothing is to be dropped off and if it is an emergency item she needs to contact DH and coordinate. 

No way should you ever be held hostage in your own home waiting on a crazy person to show up. 

As far as the questions - I would simply explain major family decisions will be discussed as a family. Everything else is adult business and never to be questioned, period. Next time it happens issue consequences. 

Say to SS:  "The things most people want to know about are usually none of their business"

Lather, Rinse, Repeat

I could see BM bringing some items that maybe were not washed and ready yet when Dad picked up the kids. Yes, of course, BM needs to text and consult about convenience of Dad about time. And with your BM, she may have told her little darlings that she will make a weekly drop of all those yummy forbidden treats along with lunch of nuggets and fries. 

Who knows what that BM promised to drop by. I'd want an adult home to receive at the door whatever it is. 

Now this one:

"  I asked DH why he even talks about this kind of stuff in front of SS. He said that he won't anymore. Then later on SS was trying to ask me questions about what we were looking at on DH's computer. I just told him it was none of this business. "

You are correct, Dad shouldn't be doing this stuff or talking about it while the boys are around. The man has had all day , day after day with no one home to have done these things. Evening after evening with no kid around to privately discuss job searches, potential moves, distance of driving blah blah with you..... instead he chooses to in front of the boys. If it was some new potential, he could do it after they went to bed or asked you to look it up while he took boys out for some exercise.

The boys did what (IMO) most kids would do. Popped up with a gazillion questions 'are we moving?', 'are you moving?, 'Austin? What's in Austin?'

Of course they are curious. How will it affect them? Will it effect them? And by sending them off and telling them to mind their own business (when the 'business' was done right in front of them) is making it a big mystery secret. Just going to make them more curious. 

I assume, as the children know Dad is currently unemployed? If not, they will today when Dad fails to go off to work. IMO , a simple 'Dad is job hunting so he can support his kids and pay his bills'  is direct , short and honest. Job loss and finding re-employment is something that goes on in many households, intact families as well as step families. Housing, food, clothing, electric bills. It all cost money and there is no money tree outside the window. 

These two boys aren't that little anymore. They are old enough to learn about life isn't free and parents must work. Adult responsibilities and all that jazz. As kids, it's not their place to have to worry about finances and all, just that belts will be tightened as necessary and a 'this too shall pass'. Perfect example is how tightening belts is why there won't be lots of out of the home entertainment this visit, bills need to be paid. 

I wouldn't treat this any differently than any intact family deals with unexpected sudden unemployment. The parent tightens the budget, looks for new employment and prepares to be able to work where the jobs actually are.

Don't forget to search any arriving deliveries from BM, who knows what she will try to smuggle in.

It's going to be a long month, hang-in there. Remember to hide from them all when it's gets to be too much. Drinks with your co-workers after work, a latte in the air cool cafe before heading home. Just make time for you, where ever and however you need to do it. 

Some interesting stories to "overhear."  How you've won the lottery,  how you've been selected to be the next woman in space/on the space station,  how your great grandfather invented Microsoft Windows. . .etc etc.

Oh yes! This is a great opportunity to start talking about vacations to Disney and Hawaii and getting a new car. 

As for BM bringing stuff to your house, I don’t know how to get around that without seeming like a crazy person. I certainly wouldn’t wait for her but if she comes and you’re home you have to answer the door.

Maybe watch for her & send SS out to the car before she can get to the door? Or let SS answer the door with DH standing behind him blocking her view? Get the stuff, say bye and slam the door?

Send her a Trespass Warning Letter by certified mail, and copy your local PD.  You can find samples online, and fix the wording to fit your situation.  Like she needs to only come during scheduled exchanges, and she must stay in her car, etc.  Then if she deviates, call the PD.