If we filed Chapter 7 during the 1 week period between the court and eviction dates, will we be allowed to remain in the residence?

To avoid an eviction, you have to pay the back rent. So few people use bankruptcy to fight an eviction.

You would be wise to meet with an experienced BK lawyer. Filing BK normally does not stop an eviction proceeding. There may be an exception.

The eviction will be stayed but you will not necessarily have the right to pay the rent and remain since you have already violated the agreement to pay the past due amount. The landlord will have the right to file a Motion to Lift the Stay to be allowed to proceed with the eviction if they want you out. A Chapter 7 might buy you a few extra days or weeks but it will not give you the right to stay very long over the landlord's objections.

Probably not. One of the sections of the bankruptcy code gives you up to 60 days after the date you file the bankruptcy petition to cure any default. So to the extent that may apply you could stay somewhat longer. There is not universal agreement among the lawyers or courts about this matter though. You certainly would benefit by retaining a skilled bankruptcy lawyer. It's almost always worth the investment.