If there is an agreement between the competitors that each one of them pays an entry fee, in addition to a condition that the winner will receive a prize, then this agreement is prohibited. If the agreement was to pay a specific amount of money to the organizing committee for participating in the tournament, and the organizing committee has control over the collected monies in managing the game, without putting a condition that obligates the committee to give a prize to the winner(s), but the committee gives the prize as a gift without any obligation, then the agreement in this form is permissible.
It is not obligatory to remove the dirt under the nail, since this area is not part of wudhu or ghusl. If the dirt prevents water from reaching the skin in parts of wudhu and ghusl, it must be removed; one can use material that removes such substances in the conventional way, and one need not focus on this beyond that. However, if a person has a reasonable doubt whether there is any substance that is forming a barrier between the water and the skin, then he should remove it to ensure that water reaches the skin.