Google Now + Evernote. Well, their 23rd century FUP equivalents.
You are dealing with a super-powerful computer, for which recoding (video, audio and other sensors) everything that happened on the ship is trivial.
All that the Captain needs to do is pick the important pieces of the recordings and, upon reviewing them after the events (hence, past tense), add captions/thoughts.
To address your Q discrepancy, I am firmly convinced that reciting the log "as things happened" was merely a break-the-fourth-wall artistic license, to help with the narrative. The past tense is the best indicator of it, as you noted.
Please note that Wikipedia agrees: ("Use in fiction"), though with as much lack of citing as I do above:
- The Hornblower series mentions logs to explain plot development, or to make the story more realistic.
- Reading a log can make a dramatic explanation of a mysterious disaster in most sci-fi.
- In Star Trek the Captain's log, a form of ship's log, is used to fill in the audience as to the events in progress, and acts as a more realistic form of soliloquy.
Memory Alpha confirms the fourth wall point:
... it should be noted that in some of the first episodes of the original series, Kirk's log sometimes broke the fourth wall and described information that neither he nor the crew could be aware of.
For example, in "The Naked Time", Kirk describes "... but unknown to us...".
There clearly wasn't time to make some other log entries; for instance, in "By Any Other Name", Kirk appears to have made a log entry about a his crew's skulduggery against the episode's villain while he, Kirk, is sitting next to that same villain. (Perhaps Kirk recorded the log entry during the commercial break!)
Presumably, these log entries were created after the events had transpired, when the Captain had time to update his log, and it was either Starfleet procedure or just Kirk's personal habit to record such logs in the present tense.