A GP is a partner and as such has an financial interest in the success of the fund. In fact, they have a greater financial exposure than LPs, since they have unlimited liability for the partnership’s obligations (how much upside they have depends on the partnership agreement, its possible that LPs have all of that and a GP that's not also an LP would not — but just having downside exposure is still a serious financial interest.)
1. Why do you assume he wouldn't receive carry? Carry is standard so the presumption should be that he receives carry unless otherwise clarified.
2. GP commitments are also common. LPs want GPs to have "skin in the game" and I think it's absolutely reasonable to claim the GP "owns" the fund (just not as a sole owner).