Christmas was good this year and I got a few wizard 101 time cards. I cashed them in and now need to choose what to do with the pets. It's my first time through the game so I have only one character and on account. If accept all three pets into this character (myth/life) and I make an alt on the same account some time later (say life/storm or such), can I trade the unused pets or matured pets to the alt? What are the restrictions (besides not auctioning them) on how I use the trained and untrained pets?
As Professor Greyrose mentioned, almost all of the pets can be shared between characters on the same account. One thing you should be aware of with time card pets, is that exactly what they can do changes from month to month. If you look at the main page here - https://www.wizard101.com/game/giftcards - the link for each card shows what the pets will be for the next few months. If you just want the pet for the looks, it doesn't matter much, but if you wanted the Kraken spell for your new Storm wizard, it can make a big difference.
Keep in mind, which month you accept your card pet determines what spell card the pet gives you. If you accept all of your pets on the same month they will all give the same spell card. Edit: I think all of the ten dollar card pets give the same spells on the same months the twenty and twenty-five dollar card pets may give different spells from the ten dollar ones. Some months are great and gift card pets accepted on those months allow you to cast one dragon blade or driad spell, other months are not so great and when you accept the pet your pet allows you to cast one storm shield or storm prism. I do not like those storm spells on pets, so I avoid accepting my pets on those months. There is a page somewhere that tells you which spell your gift card pet will provide if you accept it on which month. I don't remember where it is, but it lists all of the months, pets, and the spells they give you.
