it means that you need a pet that has a total of 100 stats of health, strength, defense and speed combined. you can have 50 health, 20 strength, 15 defense and 15 speed for example. any combination that adds to 100 stats should work. you can train your pet here. there are other ways to raise your pet's stats, but if you only want it to enter the blue building then just using the training center is the best way.
Or there's sometimes the option of finding a pet in the pound with decent stats... assuming you think training takes too long. But that can be difficult.
That's really tricky, I used to think I needed a pet rank 100 lol (between the 100 strongest pets on Subeta). But as it was said, only need to have 100 on overall status.
You can train your pets faster using boosters. They're expensive, but you can always get one by doing Shinwa's Quest (once a day, and it's not guaranteed you'll get a booster everyday). Also, some mage amulets that are also boosters can be found randomly through random achievements.
Oh, and if you battle just a little bit, especially with low status pets, you'll get amounts of EXP. You can spend them on the Battle Coliseum (HERE). According with your status, the "price" to expend will be higher, so yep, it's a really handy feature for beginners, and most of us commonly just find about it late. Guide with weapons by tiers: Here More about tiers and battling: Here