1. Your maximum is your maximum. If you enter combat with 3 stamina out of a max of 5, then your max is five. When Cthulhu attacks, you may reduce this to 3 out of 4. Then 3 out of 3. Then 2 out of 2, since your current can't be higher than your maximum. Of course, on any given attack, you could always choose to lower your stamina max instead of your sanity max. In general, investigators die on Cthulhu's 7th attack.
1. So, let's say than an investigator whose card showed a maximum Sanity of 5 entered combat with a current Sanity of 3, then the sequence would be like this:
Start of combat: Sanity 5
After 1rst Cthulhu attack: Sanity 4
After 2nd: 3
After 3rd: 2
After 4rth: 1
After 5th: 0
(For simplicity reasons I left Stamina out)
2. If Duke was there, then the investigator would start with a Sanity of 6. Right?
3. In the above example I discarded the "Dreams of Madness" effect which would reduce the investigator's Sanity of 5 to 4 at the beginning of the game, since this should be in effect only during Cthulhu's sleep (the first phase of the game) and not after Cthulhu's awakening. Was I right?
If Duke was present, in the above example his optimal use would be to keep him till I have 1 Sanity left, then to discard him to restore my Sanity to 5 (not 6, since the Sanity would be increased to 6 only while Duke was in play). Right?
Last question: If during that combat, I expended a sanity point to cast a spell, this wouldn't affect my maximum sanity, just my current one, so, practically, it wouldn't make any difference (contrarily to a battle vs. Hastur or Yig where the expenditure would hurt). Right?
2. he would start with maximum sanity 6, current would still be 3.
3. no, his slumber ability stays active. it's rather his imminent presence causing the effect.
if duke was present you should not discard him for the heal, as this will make you have another step lower maximum sanity, which is equal to one attack from cthulhu. having 6 maximum stamina going down to 1 obviously takes longer time than if you have only 5 (as with the +1 sanity from duke).
casting a spell would not hurt you no, and eventually your maximum sanity will catch up with your current sanity.
an investigator with 1 of 5 stamina and 1 of 5 sanity lasts just as long as one with 5 of 5 stamina and 5 of 5 sanity.