So, is there a relation between 'carry', 'door', and 'seaport'?

Yes, in Latin, porta 'gate; door', porto 'to carry', and portus 'entrance; harbor'] are all related. They are from PIE *per-, *perə- 'to transport, carry over; to come across; to translate; to pentrate; to fly'. Congeners in other IE languages are German fahren 'to drive, ride, travel', English fare, ferry, English ford, Greek Ευφράτης (Euphratēs) 'Euphrates', literally 'good ford', by folk etymology from Akkadian Purattu, cf. Sumerian Buranun.

Ceci n'est pas un seing.