"Don't look a gift horse in the mouth" (Don't be picky about gifts/help)
One mistake: it should be "at" instead of "a".

I don't think there's a Chinese idiom exactly equivalent to this.
You can say "别人给你礼物/帮助，你就别挑三拣四的了。"

"A bird in the hand is worth 2 in the bush" (Crazy English @_@ ) (A sure bet is better than a risky higher yield)

I would like to say "（你就）知足吧，别太贪了" to advice someone not to risk for more than he already has.

"it's all just water under the bridge" (forget about the bad relationship we had in the past)
Similarly, you can say "过去的事，就让它过去吧。"
That literally means let go the bad things (not only bad relationship) that happened before and restart for a better future.

"better late than never"
Exactly, "亡羊补牢".
That literally means: After a wolf came into your sheepfold through a gap in the wall and killed some sheep you breed, just fix the gap to prevent things go worse.

"a chain is only as strong as its weakest link"
Exactly, "木桶原理".
That literally means "Wooden Bucket Principal", which is quite famous in China.
Wooden Bucket Principal means the maximum height of water that a wooden bucket can contain is the minimum height of all the wooden pieces of that bucket.

"a little birdie told me..."
"我听说……" or "有小道消息说……"

"a fool and his money are soon parted"
No equivalent saying in Chinese.
You can say "傻瓜守不住钱。"

"a picture is worth a thousand words"
No exactly equivalent saying in Chinese.
"图画(or, 图像/画面)胜过千言万语"

"a watched kettle never boils"
"欲速则不达" (haste makes waste)

"Achilles's Heel"
No equivalent saying in Chinese.
You can say "唯一的缺点" or "唯一的致命弱点".

"All's fair in love and war"
I suggest "情场如战场，都要不择手段。"

"Don't look a gift horse in the mouth" (Don't be picky about gifts/help)

守信用

"A bird in the hand is worth 2 in the bush" (Crazy English @_@ ) (A sure bet is better than a risky higher yield)
聊胜于无

"it's all just water under the bridge" (forget about the bad relationship we had in the past)
忘掉不愉快

"better late than never"
亡羊补牢

"a chain is only as strong as its weakest link"
木桶原理

"a little birdie told me..."
小鸟说

"a fool and his money are soon parted"
不理财，财不理

"a picture is worth a thousand words"
图画的变现力胜于言语

"a watched kettle never boils"
踏破铁鞋无觅处

"Achilles's Heel"

致命弱点

"All's fair in love and war"
爱情与战争无公义可言