1. What are advancement points?
2. How long will it take for my starter to grow to adult?
3. How long will it take for my hatchlings to grow?
4.Why did I randomly get a experience orb?

1. What are advancement points?
Points you get for playing the game, doing various things (breeding, nurturing, battling, doing missions etc.
Currently they are used for three things only- unlocking base styles in the wardrobe, buying more save slots in the generator, and buying more party preset slots on the party page.
Krin's intent is that in the future there will be an AP shop that has things to buy with them.

2. How long will it take for my starter to grow to adult?
All starters start life at the beginning of adolescence- that is, 216 maturity hours (the same as any pet made via the generator). This is the equivalent of 9 days old.
Pets reach adulthood at 648 hours (equivalent to 27 days old); adolescence stage lasts 18 days.
With starters, generated pets, and pets that have been genetested, you can see their exact number of Maturity Hours next to their Maturity descriptor- currently (as of me typing this) your starter TidalFin is "Maturity: Adolescent (309/648)"; with some quick maths, 648-309 = 339.
339 hours is just a little over 14 days.

3. How long will it take for my hatchlings to grow?
Egglings take 72 hours (3 days) to hatch from the moment of being 'born' (eggs found while exploring can be any amount progressed)- you can see their maturity hour counter in their profile (when they hatch it's replaced by their hunger meter)
Every hour adds one to the counter, until it hits 72. For every six correct nurtures that an eggling gets, it gets a bonus +1 to the counter

Hatchling stage lasts from 72 hours-215 hours of age (with, as mentioned, 216 being the start of adolescence.) (meaning it lasts 6 days)
They get +1 bonus to their maturity hour count for every four correct nurtures they get (this is to counterbalance the bonus stat they get every 8 correct nurtures)

4.Why did I randomly get a experience orb?
exp orbs are just a random drop you can get from any enemy. They'll always drop an exp orb matching their level. You got one because the rng rolled it from the drop table.

Another question: How to get diamonds?

The most reliable ways are by buying them in the diamond shop (with irl money) or in the curencie exchange (with gold). Some events both officials and non-oficials may also offer some as a prize or you may be able to sale extra-rare pets (wild tagged neph, high grove stats ...) /items for some.