When I was a child, I loved to play with slingshots.

Although I didn't shoot down a fewny birds, slingshots brought a lot of joy to my childhood.

I remember my father once said that he could get a home-made gun for me, but he didn't do that never ended up doing the endat.

Although I didn't shoot down a few birds, slingshots brought a lot of joy to my childhood.

I'm not sure how "I didn't shoot down a few birds" fits in.

Slingshots

The plural from is more natural as a one-word title, especially since the body of your post describes the (many) slingshots of your childhood, not just one slingshot.

When I was a child, I loved to play with slingshots.

“Slingshot” is not the name of a game or a musical instrument, so you need “play with” instead of just “play.”

At that time, China [hadn't controlled | hadn’t started to control | didn’t control] guns so / as strictly as now, so gasair guns could be bought in some shops.

We call them air guns, even though some of them do use a compressed gas such as carbon dioxide.

I only had slingshots [made by myself | I made myself | homemade slingshots].

“By myself” emphasizes that nobody helped you make them. This isn’t incorrect, but it would be more natural to simply say you made them yourself (without “by”) or to describe them as “homemade.”

When I made a slingshot, I'd use a y-shaped tree branches, rubber bands and an oval piece of leather.

Although I [didn't shoot down a few] birds, slingshots brought a lot of joy to my childhood.

“Didn’t shoot down a few” isn’t incorrect, but it’s a form of double-negative (“not a few”), so it’s ambiguous: “not a few” usually means “not just a few,” i.e., “many,” although it can be literally interpreted as “less than a few.” I understand “didn’t shoot down a few” to mean you shot very many birds, but I’m not sure this is what you intended to say.