Yes. You cannot go to any excessive means to obtain hidden information but if your opponent happens to accidentally reveal something, you are most certainly allowed to use the information.
"Must be at the beginning"
"Do not need to be at the beginning"
^ That is a contradiction, not an expansion. An expansion would be adding optional, additional rules that can coexist with the main ones. That's not what's going on there. These two things cannot both be true at the same time.
The Rules Theory and Templating Forum is likely a better venue for your concerns. I suggest making a thread there if you consider this an issue.
Edit: The word 'Must' does not occur in rule 603.1, I'm failing to see your point.
Delayed triggered abilities are a subset of Triggered abilities, hence why they are described later in section 603 of the CR.
The rules do not contradict, they are merely expanding on a single concept.
Your argument on replacement effects is not valid since delayed triggered abilities still must contain one of those three words, the only difference is the placement of said words.
During the declare blockers phase how would pit fight interact?
Can I use it on an attacking creature to have it eliminate another creature that is not blocking it and (if it survives) still do damage to the blocking creature?
Yes, fight does not remove creatures from combat.
Willl damage received from pit fight stay on my creature when determining damaged from blocking creature?
Yes, damage remains marked until the Cleanup Step at the end of a turn.
Opponent ends his turn. During end step, I cast Brainstorm. Asks for response. He has no response and passes priority back to me. I then cast my Notion Thief.
You do not get priority back after your opponent if he does not respond. Brainstorm would resolve and you do not have an opportunity to cast Notion Thief. If you want to cast Notion Thief you would need to hold priority after casting Brainstorm or do so before casting Brainstorm.
To answer the question you are asking though, yes you would get to draw three. If both players in a game control Notion Thief they effectively cancel each other out.
At the point during the process of casting a spell where you need to choose targets you already chosen to pay the Bestow alternate cost which means the bestow card is no longer a creature spell.
Say you play Zealous Conscripts and you target Olivia Voldaren. Then you get back some creature Olivia bit as well. I assume your opponent doesn't get control of your old creature back once you return Olivia right?
Correct, the effect ended when its duration was met (no longer controlling Olivia).
I'm also assuming they don't have to bite that creature again just use the bottom ability by itself if they would want to take it back again, but just wanted to double check.
Correct, that creature is still a vampire and can be targeted by Olivia's third ability.
Its too late, the trigger will have resolved during the declare attackers step which is when it checks the power of the Beastmaster. It does not continually check.
Upside, you draw 7 cards.
The Rules Theory and Templating Forum is likely a better venue for your concerns. I suggest making a thread there if you consider this an issue.
Edit: The word 'Must' does not occur in rule 603.1, I'm failing to see your point.
The rules do not contradict, they are merely expanding on a single concept.
Your argument on replacement effects is not valid since delayed triggered abilities still must contain one of those three words, the only difference is the placement of said words.
Yes, fight does not remove creatures from combat.
Yes, damage remains marked until the Cleanup Step at the end of a turn.
You do not get priority back after your opponent if he does not respond. Brainstorm would resolve and you do not have an opportunity to cast Notion Thief. If you want to cast Notion Thief you would need to hold priority after casting Brainstorm or do so before casting Brainstorm.
To answer the question you are asking though, yes you would get to draw three. If both players in a game control Notion Thief they effectively cancel each other out.
At the point during the process of casting a spell where you need to choose targets you already chosen to pay the Bestow alternate cost which means the bestow card is no longer a creature spell.
Correct, the effect ended when its duration was met (no longer controlling Olivia).
Correct, that creature is still a vampire and can be targeted by Olivia's third ability.
(I believe the recommended is somewhere in the 30-40 range)
Infect simply modifies what effect damage has, if the damage never gets dealt than infect does nothing.