Is it OK for my boyfriend to help me around with chores? (my responsibility)
Yes, It is ok, as long as he is helping you and not doing it for you. If you are running late or are about to go out and you are held back by a couple of chores, YES, you can ask for a little help. And if he's willing to help, then why not???
But, if he doesn't want to help, that's where it is wrong for you to be pushy and needy demanding for his help. Your chores are your chores, not his. If he doesn't want to help, there is nothing you can do about it and He wouldn't be wrong. This even applies for marriage!
I'm going to give you an example of my husband and I. My husband helps me when I need help; Almost ALL the time. That is a character trait I love about him. It comes naturally from him, whether I ask him to help me or not. Sometimes, he sees me struggling to do or finish something and he offers to help on his own. It doesn't matter what it is. Just between you and me....My husband can throw it down in house cleaning! If we have to move and I'm running up and down getting things ready for the move. My husband almost always starts helping me to clean the house.
My sisters husband on the other hand, doesn't help clean the house when they are moving, but he helps by completely packing his own suitcase. That's one less thing she has to worry about.
I also only know one other pastor that helps his wife. This Pastor/husband would help his wife in anything needed. And never once did I see the wife asking him to help. This desire to help her would come from him. They had a small child, and this pastor would help his wife with things like cleaning, cooking up something simple for them to eat, it was small things, but things that count.
Finding a man that likes to help his woman is VERY RARE. Don't get me wrong. This isn't the husbands job, and not every husband/man does it. Some husbands don't help the wife at all with their chores. And that's OK according to the Bible. The Bible says that all types of house work and to care for the family is a wives job. That's why we can't impose on our man to do our job. But some men like to help their woman, and to me I wouldn't bust his bubble.