Two possessive nouns in a row

Yes, the first sentence is correct. Although I suspect parent's may be intended as plural here, in which case the apostrophe should be moved to be parents'. I suspect this because if the house belonged to one parent, you'd be more likely to write Lauren's mom's/dad's house.

"I am going over to Lauren's parents' house this evening."

The next two sentences can actually be simplified by removing one of the possessives.

"The store manager's niece was quite attractive."

Store manager is a title, so you don't have to use a possessive to describe the store's manager. The same goes for the army general:

"The US Army general's wife's dog is on the lamb. Somebody stop it."

(You also had a comma splice in there, which I've fixed.)

Disregarding the improvements to the second and third examples, the way you have used the multiple possessives is correct. However, it is best practice to try to reduce the number of consecutive possessives in a sentence if you can.