I usually zap with Phoenix but I receive zaps with Zebedee.

the fees are for setting up channels which the non custodial wallets need to do. You can turn off on the fly channel creation but then some payments might fail. These are some of the "pain points" or bad UX of the non-custodial wallets. There is a lot more to know and a lot more to do and cost, but this comes at increased sovereignty.