When you have kids and pets?
Eviction laws are different in each state, but generally, the landlord needs to follow a process to the letter, in order to carry though the eviction properly. In most states, you can be evicted with no reason, with enough notice given. If the landlord needs a good reason, eg. nonpayment of rent of breaking the rules of the contract... and if those problems are corrected within a certain number of days, could be 30 or 60 depending on the State, then the eviction notice is cancelled. I dont think winter, pets or kids have anything to do with it.
Unless your landlord has a 'no pets' rule, it should be fine. Also, depending on the contract and even state laws, the grace period given to you to move out is different. You have to go through the contract you made with your landlord to find out. I don't think there has ever been a 'no kids' rule.
Yes, with proper notice.
Check the Landlord/Tennant Laws in your state or country.
I hope you do not take offense, To that that's gone before; 'Tis only that it's my two-cents, And not one penny more. §;o)