I generally write in Third person, present tense.
I will write in Third person, past tense if the DM has put up a wall of text covering a few days / hours / events for the players to write their responses to.
I'm in one game where there are HUGE amounts of walls of text as the DM is very trusting of his players and will often post a full day of events leaving us to write responses from NPC's / modify the way things happened slightly / interact with NPC's we bring into the game / etc
I think third person is easier for others to read, and I understand that by writing in third person, the Player puts themselves in a story telling role about their character.
In tabletop games I use First person, present tense as I am that character at the table.
I have played a game on MW in First person, present tense as it was Geo-political, with the players not actually meeting -- and it felt completely natural RPing in First Person as it was mainly RP'ing, not much combat to worry about.
As for thought processe, they are all first person, present tense -- though serious "thinking" is in private text for the DM only.