All the examples I can find in dictionaries use “make a promise”, but people also use “give a promise” and there is no difference in meaning in my opinion. Here are two examples with “give a promise” from The Guardian: When Seb Coe said 'I promise' he didn't promise for himself, he promised for you, he promised for me. He gave a promise on behalf of us all." (https://www.theguardian.com/uk/2011/jan/27/tottenham-west-ham-olympic-stadium) Trimble gave a promise to his party's rank and file last November to resign as First Minister if there was no progress on decommissioning by February. (https://www.theguardian.com/uk/2000/jan/23/northernireland.politics1)

As far as I know, Make promise is the correct one.

Yes to give an offer anything either promise is section must be implemented