I like watching the birds comingthat come to my garden.¶
OR ¶
I like watching birds come to my garden.

In winter, I can often see white-eyes and bulbuls in a camellia tree.

Bulbuls drink honey so much andare birds that drink a lot of honey, and they sometimes attack white-eyes.

This sentence sounds more like you are teaching us about bulbuls. Otherwise, mentioning the fact that they drink so much honey might feel unnecessary.

So, every winter, I set up a special cage whichthat I ordered from a bamboo artisan.¶
OR¶
So, every winter I set up a special cage that I ordered from a bamboo artisan.

You can use "which", but "that" feels a little more natural to me.

It is like a globe woven by bamboo at random roughlyroughly and randomly woven bamboo globe. ¶
OR¶
It is kind of a bamboo globe, woven roughly and seemingly at random.¶
OR ¶
It is sort of a globe-shape, roughly woven at random with bamboo.¶
OR¶
It looks like a globe, roughly and randomly woven with bamboo.

"Woven by bamboo" sounds like the bamboo is the weaver. :)
"Seemingly at random" is a phrase I put here, since the bamboo artisan probably wasn't actually weaving randomly. It would be difficult to work that way, right? But it looks random, so: "seemingly at random". It isn't necessary though, it is up to your preference.

There are many different sized wholes on the surface, but every size of the wholes isare all bigger than the size of a white-eye and smaller than the size of a bulbul.a bulbul. ¶
OR¶
There are many different sized holes on the surface, but the holes are all too small to let the bulbuls inside.

The second sentence sounds more natural to me. The size doesn't need to be specified this much.

So only white-eyes can eat the oranges I hang inside the cage.

I like watching birds cominge to my garden.¶

Bulbuls drink honey so much honey and sometimes attack white-eyes.

So every winter I set a special cage which I ordered from a bamboo artisan every winter.

I like watching birds cominge to my garden.

In the winter, I can often see white-eyes and bulbuls in amy camellia tree.

I corrected this assuming you're talking about a camellia tree in your carden.

Bulbuls drink honey so much honey and sometimes attack white-eyes.

So every winter I set a special cage, which I ordered from a bamboo artisan every winter.

It is like a globe woven by bamboo at random roughlyroughly and randomly with bamboo.

There are many different sized wholes on the surface, but every size of the wach holes is bigger than the size of a white-eye and smaller than the size of a bulbul.

So only white-eyes can eat the oranges I hang inside the cage.

I like watching birds cominge to my garden.

InDuring the winter, I can often see white-eyes and bulbuls in a camellia tree.

Bulbuls drink honey so mucha lot of honey, and sometimes attack white-eyes.

So every winter I set a special cage, which I ordered from a bamboo artisan every winter.

It is like a globe woven bywith bamboo atroughly and random roughly.

There are many different sized wholes on the surface, but everythe size of the wholes isare bigger than the size of a white-eye, and smaller than the size of a bulbul.