Strictly speaking 'Ga i' is only 'May I' but because English, especially UK English, very often replaces the correct 'May I' with 'Can I' in speech when asking for something we've included that as an alternative translation.

The distinction is there in Welsh as well. Among its several meanings, cael can be used, as here, to mean 'being allowed to'. So ga i ...? is used as the equivalent of 'may I ...?'

'Can he swim?', as in his ability, would be ydy e'n gallu/medru nofio?.

The t in talu (paying) is soft mutated to the d in dalu here because talu is the object of the verb ga (may I...?).

This is explained with more examples in the notes for the section 'May I?'

Same as in English, really. If you specifically want the bill to look over before you pay, you would ask Ga i'r bil?.