Make sure you've completed all the marriage requirements. Which are:
Have a Red Heart with her
Complete her Heart Events
Upgrade your house twice
Buy the Wife's Bed
Unlock all 23 main islanders
Also Nathan and Alisa must currently be living on the island. They are the ones who preform the marriage ceremony so if they are not there you can not get married. Even if you have completed all the requirements no one will accept your proposal until Nathan and Alisa move back.
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To give gifts to people, you have to find out what they like or love first and then get that present (whatever it is) and just take it out of your rucksack and talk to them and they will accept it.
Yes, It is necessary. You will need to find at least 60 of them to get married. Before that, the girl/boy you are dating won't accept the Blue Feather.
You propose to him using the Blue Feather. You can buy the Blue Feather from Karen's Supermarket once any marriage candidate has an Orange Heart. But he will not accept your Blue Feather until you complete all the marriage requirements. Which are: Upgrade your house twice Buy the double bed Have a Red Heart with the guy See all of his Heart Events Rescue at least 60 Harvest Sprites
You need the Blue Feather in order to propose. When you have that equip it in the tool slot, then stand next to the girl, use the Blue Feather like a tool to show it to the girl you want to propose to. She will only accept if you've completed all the marriage requirements.