Hey there @Celozon, welcome to the site!

- From the sound of it, RNG just isn't being nice to you. Which is a bummer because I'm sure you spent a fortune on those egg potions. I'm brewing up a bunch of egg potions right now, so I'll send some your way in about an hour :)

- In order to export items out of the adventure, first you'll have to make sure that the pet you want to take items from is on your adventure squad, but has finished their adventure. If your pet is still in the adventure, you'll want to take them back to the Entrance Gate in Gateway City and click "Finish Adventure" in the bottom right. This will place them on standby and save their progress. Then you can export items by clicking the "Inventory" button next to the pet's sprite. On the next page you can select an item, then click "To Main Item Inventory," then select the quantity, and finally click "Send." (It's almost the same instructions for importing items to the adventure, except you'll want to click the "Equip" button next to the pet's sprite.)

- "Finish Adventure" simply puts the pet on standby so you can import or export items from them, it won't erase their save or anything fortunately!

- While you can have multiple pets in your adventure squad, you can only have one pet in the adventure at a time. Other pets in the squad will be on standby until the current pet finishes their adventure.

- As for the time of day affecting things other than quests, I'm fairly certain it does affect some items like Strange Coins and Jade Shards (which can be exchanged for backpack upgrades), but I haven't played the adventure in a long time so I wouldn't be able to provide a list of exact times, sadly xwx

Please let me know if I missed any of your questions or if anything I said was confusing. And don't hesitate to ask more, the community is happy to help!

Also hope you have better luck with capturing :)