"tip" has at least the two meanings "a gratuity" and (informal) "a useful piece of advice". I wouldn't use "hint" here.

I'd use:

I recommend ...

The two words are very different. A tip is when you tell a person useful information, to help them. A hint is when you don't actually tell them, you say something different which might help them find that information themselves.

e.g.

A: How do I delete this word?
B: I'll give you a tip: the delete button is on the top right of your keyboard.

A: How do I delete this word?
B: I'll give you a hint: look on the top right of your keyboard.

Oh, and a tip as in Trinkgeld is not related to this meaning of "tip" and has no effect on how often this meaning of "tip" is used.

Thanks.
So ... "tip" is used if you give an advise for a mandatory action and "hint" is just for additional information?

No, tip is when you give some clear advice, hint is when you hint at something, i.e. you give them a clue but you don't say it explicitly.

#4 No; in "give s.o. a hint", a hint is when you say something indirectly, as in "It's a bit cold, isn't it?" when you want someone to close the window, or "You look very smart" when you mean someone is overdressed.

a hint is a clue

a tip is a bit of helpful information

2:45 Newmarket Zero Tolerance

My tip for today.

Anyone know anything about this funny word "cumshaw" by the way? related discussion: cumshaw

tip ist ja auch auf Deutsch ein Tip, hint ist ein (oft versteckter) Hinweis.
beim obigen Beispiel mit dem offenen Fenster:
"It's a bit cold, isn't it?"
wenn nun der/die andere nicht reagiert, könnte eine dritte Person sagen: that was a hint (to close the window)

It was probably British Navy personnel who first picked up "cumshaw" in Chinese ports, during the First Opium War of 1839–42. "Cumshaw" is from a word that means "grateful thanks" in the dialect of Xiamen, a port in southeast China.

Here's the etymology of "cumshaw" from the OED:

According to Giles, the Amoy pronunciation, kam-sia, of the Chinese words kan to be grateful, hsieh thanks = ‘grateful thanks’, a phrase of thanks used by beggars.

Does it mark it as rare or archaic today? Is it still used anywhere?

No, it doesn't mark it as rare or archaic. To be honest I hadn't heard of it before reading this thread ;-)

Here's the full OED entry:

In the Chinese ports: A present or gratuity; a baksheesh.
1839 H. Malcom Trav. II. Gloss., Cum-shaw, a present. At Canton, custom has made some cumshaws matter of right. 1885 Where Chineses Drive 163 Baldpate ... had the exceeding coolness to ask for a cumshaw as they left.

Hence cumshaw v., to make a present to.