I am very new to GOW and am learning new things every day... and discovering more things to be curious about each day, too...
I finally noticed that each player only gets a limited number teams to use.... Since there are more situations requiring unique teams than the number of team slots, how do you save teams you want to use again, and how to you later, recover those teams to reuse as needed...
do you use some function of GOW or do you need to use a spreadsheet, or some other external program?
First of all, to the site! I've moved your topic to the Gems of War area, but you should be able to find it easily.
In answer to your question, the higher we get in levels, the more team slots we will be awarded. It's one of the perks of leveling your hero. There are other ways to get team slots, such as purchasing Warbands and increasing your VIP level (VIP costs real money, though).
As for how I keep from losing teams, just take a look around this GoW area (scroll up to the top of this topic and click on the Gems of War link). I have teams posted everywhere, because I have WAY more teams than team slots!
Pet rescues and special events have their own team slots that are for the event only so that you don't have to use your limited team slots for them.