Unfortunately, no.
You can get Falabellas on howrse by buying them if anyone is selling them and promotions there is one currently running called wandering horses when you get all the horses you get a balios witch is for the first 10 players to get all the horses and for the rest of the people they get a Falabellas
Falabellas are anywhere from 26" to 38" in height at maturity.
You can no longer get them for your ECs: they were removed along with illnesses a few months ago. They have now been 'remodelled' and re-released as a Special Howrse, with 30 GP and hardly any skills. They offer a 1 in 100 chance at comp bonuses for owners. You can get one in a horn of lushness if you're lucky. Sorry, but no, you cannot. Falabellas are no longer available for ECs, period. The only way you can get one of the new Special Pegasus Falabellas is by buying it from another player, if anyone is selling.
Sleipnir can only reproduce when another Sleipnir disappears.
No, Archimedes is a Special horse and therefore cannot reproduce.
Horses can reproduce throughout their lifetime.
There is only one Cowbra; it cannot reproduce or be sold. The owner's name is Michiba.
around £700 but they are cute
No, it cannot. All Sleipnirs are male, and the majority of the Divines do not have the ability to reproduce anyhow.
it depends what you are doing but not normally
If you have a mare, go to the bottom of her page, look under the heading REPRODUCTION and click either private or public coverings. My name on howrse is Miss-Charli if u need a hand add me as a friend or send me a message!
It takes 8 months so 4 days or just use aging points