His back story comes up later and explains a few things (like why he and the hero party are not aware he is not a standard tamer, but a massive upgrade from the rest, cat girl does have some knowledge as can be seen in her reaction the first few times she sees him use his abilities)
He has never been part of the guild. the hero party are not members, and operate independently of the guild.
He tends to make temporary contracts with the local fauna, so would normally pick up something mid quest, if there was nothing good for combat in the immediate area why bother.
The girl offered boobies, though seriously you will see what she gives him after the arm wrestling, which is partly to give an opertunity to demonstrate what she gives him.
Muscle heads and pier pressure combined with male ego explain the arm wrestling, it is better to accept and lose then reject and be labeled a coward/ostracized. Especilly if he might need to join adventurer parties in the future or interact with other adventures.