I'd like to ask about the expression of a request.
I heard that " Will you ～?" sounded strong and it was like an order. Is that right?
I know " Would you ～?" or " Could you ～?" are more polite. And there is another way of " Will you ～?" like " Will you try this cake?"
I wonder in what situation and to whom you use " Will you ～?" for requesting.
Would you help me to understand this usage?
We could write a book on this and no one would agree and it still wouldn't always make sense :)
The usage of will/could/shall is so convoluted and largely archaic, I don't think there are anymore definite rules. As always, intonation is a FAR stronger indicator of mood and intent than anything in written text is. You can say "I love you" and make it sound like "I want to kill you" if you really wanted to.
That being said.... any of those forms are acceptable for a request.
"Will you be joining us for dinner"
"Would you care to dance?"
"Could you pass the salt?"
"Shall I order us some drinks?"
In answer to your question, you can simply ask "Would you like to try some of this cake?"
It all depends on context. If it's spoken, then it all depends on intonation.