"peux" is the je form of "pouvoir", which means "to be able to" or "to be permitted to." "How do I carry it" means "In what way or manner should I carry it," whereas "How can I carry it?" has connotations of "How can I be expected to carry it?" or "How would it be possible for me to carry it?" Anyway, that's my best guess.
Only because we don't use "can be“ like that, or say "canbe“, but we do say "could be“. In this case, as to the niggling distinction made between "can“ and "may“, "may“ is usurping its position, as prescriptivists claim "can“ is doing to "may“. The future does not ask permission, so "maybe“ might be another case of "can I go to the bathroom?“...
Puis-je is pretty old-fashioned these days. Est-ce que je peux... is more common. They mean the same thing. (It's not a different tense, just one of those things that result from the evolution of the language.) Check this out: https://www.thoughtco.com/pouvoir-can-to-be-able-may-1370664