Did the seaman pay Jim’s father for staying at the inn?

The answer is yes and no..... he paid an initial deposit, but he never paid for his entire stay. An example would be booking a week at a hotel, and then staying an additional seven months without paying for your room.

The seaman initially paid Jim's father a specific amount for staying at the inn, but when the money was exhausted.... his stay continued, and he never paid another penny.

In one way, indeed, he bade fair to ruin us, for he kept on staying week after week, and at last month after month, so that all the money had been long exhausted, and still my father never plucked up the heart to insist on having more. If ever he mentioned it, the captain blew through his nose so loudly that you might say he roared, and stared my poor father out of the room.

Yes

Yes the seaman pay Jim's father for staying at the inn

Yes!

Billy Bones or the Captain paid Jim's father four penny to stay in the inn.