Question 1: The death of Socrates is usually said to have happened in 399 BC, but I've sometimes seen a question mark after the date. Is there actually a possible uncertainty about the exact year? What sources is the date based on? Xenophon? Plato? What is the chain that connects this to the modern calendar?
Question 2: It sounds like his death happened early in the month of Skirophorion, so maybe June or July. Is this about as accurate as we can be, June/July? As far as I can tell, the Attic calendar had vaguely defined months, and there may be no way of matching a particular month in a particular year to the Gregorian calendar.
The clearest discussion I've found is in the Stanford Encyclopedia of Philosophy:
In the month of Thargelion [May-June 399 Apology] a month or two after Meletus's initial summons, Socrates's trial occurred. On the day before, the Athenians had launched a ship to Delos, dedicated to Apollo and commemorating Theseus's legendary victory over the Minotaur (Phaedo 58a-b). [...] Although the duration of the annual voyage varied with conditions, Xenophon says it took thirty-one days in 399 (Memorabilia 4.8.2); if so, Socrates lived thirty days beyond his trial, into the month of Skirophorion.
Would the ship have been launched on the 6th or 7th of Thargelion, which were the days of the Thargelia purification ritual?