Is it possible to defeat Azathoth when playing with only 1 Investigator?
Investigator count

Look at these statistics calculated from hundreds of logged plays: Eldritch Horror Statistics.

With the original reference cards, you can see the normal win rate for 1 investigator is 23% versus 66% for 4. If I play this game with four or fewer players, I always play with four investigators. I find this much more enjoyable.

The updated reference cards you've heard about are now considered a variant of the game: Revised Reference Cards. They make the game slightly easier for 1 or 5 investigators, and slightly harder for any other number (which you can see in the above statistics). But the difference between 1 and 4 investigators is still 32% vs 55%. So "easier" with 1 investigator is still harder than the "harder" 4 investigators.

Reckoning

From the rules reference, last bullet point on page 9:


  If a Monster is spawned or an investigator gains a component while investigators are resolving [reckoning] effects, they do not resolve the [reckoning] effect on that Monster or component.


So no, reckoning effects don't apply to monsters or possessions gained during the reckonings step. Remember also that the icons on mythos cards are resolved left to right, and the event text is resolved last of all - so if a gate opening or a monster surge spawns a monster with a reckoning effect, or if the event spawns such a monster or gives players possessions with reckoning effects, those effects will not trigger this round because the reckonings step has already been resolved.

Eldritch Tokens

The eldritch tokens on the omen track serve one and only one purpose with azathoth: they cause you to advance doom even if no green gates are open. If green gates are also open, the eldritch tokens cause additional doom on top of the normal amount.