So I've been trying to capture wild sylesties, not for any reason other than to have them (no quest) and it worked the very first time I did it but now it says I've captured them yet shows nothing when I look in my stalls, is this something I can fix?Am I doing something wrong? I've used a lot of gold on traps and I would like to know how to get the things I successfully capture...
I tried to release my first captured wild sylestie thinking maybe it was due to restricted inventory space, but I suppose I was mistaken (which I should've realized since there was no indication of the slots being locked) as the sylesties still did not appear in my inventory :( please help!

Also am I supposed to not be able to breed the sylestie I created when I signed up? It won't let me and I just want to make sure it's not an issue. Thank you!

Your starter pet will be able to breed when she's an adult. It can take a little while for adolescents to grow up into adults, but it will happen.

As for capturing pets... there's a common misconception that once the battle ends, you've caught the pet. Unfortunately this isn't always the case. It's possible for a pet to run away from a trap, in which case the battle ends but you don't get the pet. I'm guessing that's what's happened to you.

You're probably right, thank you so much!

1. nets are not always successful- the outcomes are a) successful capture, b) wild sylesti escapes the net and keeps fighting, and c) wild sylesti escapes and runs, ending the battle if it was the only enemy present.
If the capture is successful, it'll give you a link to their page in the info/battle log.

2. starter sylesti start as fresh adolescents (as do any pets created in the Generator via Essence). This means they start at 216 maturity hours. Pets reach Adulthood at 648 maturity hours- aka, 18 days after the start of adolescence.
They can only breed once they are adults.

3. you start with 25 stable slots, though one is taken up by your starter, and one potentially taken by the capture-a-sylesti quest pet. None are locked within a tab once you buy the tab. You can buy more Stables (500 Diamonds, 400 when a sale is on) and Tabs (each tab in any single stable that you get has a higher price. All Tab 2's cost 20k gold, all Tab 4's cost 120k gold etc.)