What world is better to use if a man would like to be married? Does he do "an offer" to his girlfriend?
"Proposal" is the noun, but in English it is quite common to use the verb "propose." Example: "Last night John proposed to Mary." Even if we haven't been given a context, it is clear that John asked Mary to marry him. It woudn't sound quite natural to say, "Last night John made a proposal to Mary" if you're trying to say it was a proposal of marriage.
Keep in mind that there are other types of "proposals" than a proposal of marriage. For example, a business proposal: John: "I'd like to buy your company." Mary: "Ok. What's your proposal?"
Actually the best one is not among your options xD The right word is "PROPOSAL", a word that you can use in different situations, not only talking about marriages.
It's also quite common to say that someone offered their hand in marriage.
Suggestion: 'Should we go for a coffee?'
Offer: 'Would you like me to carry your bag?'
Proposal: 'Will you marry me?'
Thank you! It is so interesting to know.