It is unlikely that the landlord can evict you under these circumstances but I would ask for advice from a lawyer in your area that understands local real estate laws.
As for showing, a landlord would have specific language in the lease pertaining to that and what is required. You would need to secure you dogs or be libel for any problems. If there is no written lease (month to month tenancy), your landlord needs to give you reasonable notice that he/she intends to enter the property and that is usualy about 24hrs. As such, you still need to secure your dogs. The person that gets bitten will certainly hold you responsible. I strongly advise you talk with an Attorney. Good luck.
The new owner may be happy to keep you on if you have been a good tenant - at the same rent.