OP, MN is a real hotbed of landlord haters. I'm a LL and once posted about a tenant who didn't pay rent and the responses really attacked me for having the audacity to expect my tenant to pay. He then destroyed my house leaving me thousands out of pocket and a damaged ceiling thanks to smoking drugs, I used the word 'scumbag' in a post and got berated for it 🙄 People on here always assume everyone wants to buy their own property and won't accept that some are happy renting. I'm also an accidental LL and have been told on here that I shouldn't be renting my property out (despite being in negative equity if I sold) as in holding people back getting on the property ladder. Never mind that paying two mortgages would cripple us financially, as long as I have a clear conscience about strangers, that's what's important 
However - with all due respect I think you need to brush up on your knowledge of tenant rights. You can't just "get them out quick" if they don't pay rent - the process means they could live in your property for 6-12 months rent free before you get a court order to forcibly remove them. They have to be 2 months behind on rent before it get the ball rolling. You can't just kick them out. If you tried they'd have every right to call the police, who'd rightly side with them, as it may be your property but it absolutely is their home and you need to treat it as their home.
It's not fair and LLs lose houses because tenants spend years sometimes not paying, but it's how the law is, and it's in place to stop unscrupulous LLs taking advantage of tenants.
If there's no "no pets" clause in the AST then they haven't broken any rules. However most AST's have a clause about keeping the garden and house in a reasonable condition and I'd certainly say dog shit everywhere is unreasonable.
They're really very good to allow viewings, they didn't have to agree to this so I would definitely ask them if they could clean the shit up from the garden but they certainly haven't 'snuck' a pet in so you can't challenge them on this. Also, arrive 30 minutes before the viewing to look round and check all is OK. It may mean picking up dog shit but if you want your house sold then needs must