More transports help planet invasions. The minimum time required to invade is 3 for colonies, and 5 for core worlds. If invasion time is longer than this, then adding transports would help. Careful though, as I've seen adding ships to an invading fleet halt invasions.

Invasions are always successful if given enough time. The selecting a fleet or planet should show how many turns are left.

I've never seen soldiering be of any help. I have halted invasions, then start again to see if it has any effect.