[RPG] How is childbirth handled

You have a few options:

1. The Animal Handling skill (PH p.178) usually applies to domestic animals, but would still be relevant to the lengthy and difficult process of raising a wild one.
2. The level 1 spell Animal Friendship (PH p.212) will tame a beast with Intelligence less than 4 for 24 hours if it fails a Wisdom save. This is a short term solution, but it could also be a jump-start for long-term taming.
3. The Ranger's Companion class feature (PH p.93) is the canonical way to tame a wild animal, but requires 3 levels of Ranger. It's limited to Medium creatures, but if your character were a Small race, you could ride your companion.
4. Taking the Mounted Combatant feat (PH p.168) gives you a strong metagaming argument for convincing your DM to let you have an interesting mount.

A Starting point for option #1...

Since we're in house rule territory, my suggestion would be to treat taming the wild animal as a downtime activity (PH p.187) similar to item crafting.

• A proficient artisan can create 5 gp worth of value per day of work.
• The "Mounts and Other Animals" table (PH p.157) gives market values for various species.
• Divide the market value of the mount by 5 gp to obtain how many days of successful Animal Handling rolls are required to tame it.
• The DM chooses a DC based on how appropriate the animal is for domestication.

For example, breaking a bronco and training it as a riding horse worth 75 gp would take 15 days of successful rolls at DC 10 (easy).

It's probably not realistic, but it's playable -- quick enough for a character to actually do it, and balanced to the in-game value of skilled labor.

Take good notes and remember for the players.

1. Take good notes. Don't rely on your memory for what happened. Instead, take notes during the game and then allot about an hour for going over and adding to those notes. Basically what you want to have when you're done and packing away your notes is a little kit of information that will tell you everything you need to know to keep running the game. Expect to completely forget anything you don't write down, and that will make sure that you have useful notes. Treat it like you're preparing notes for another DM to pick up where you left off!

   Then, weeks or months later when it's time to prepare for the next session, you can take those notes out and read them as the “background” for the next adventure, as if you weren't even there.

2. Remember for the players. During the game, be your players' memories. When you bring in an NPC that they previously interacted with, make that part of their reintroduction. (E.g., “A tough-looking woman with a broadsword on her hip enters the tavern. She's Meredith, the mercenary that you worked with briefly but parted on poor terms with. She hasn't noticed you and walks over to the bar…”)

   By being generous with this information, you make up for what the players are missing: immediate recall of what just happened last session.

These two things are all you really need to make a session work after a long time. Provided you have players who want to be gaming (i.e., enthusiasm waning due to long breaks isn't an additional issue), these practical steps all that's necessary to patch the time gap.