Of course this can happen, but the expectation is that all the players help contribute to finding solutions to the current problems. While you might see one path, someone else might see another, that happens to be better. Having multiple people working on the same problem from different perspectives generally provides better results. Kind of like crowd ...
Yes you must always tell the truth about cards in Hanabi. When you give information you give it about the card. If you give color you must give it for all cards of that color. If you give it about the number you must give it about all cards with that number. Given those two facts it is absolutely against the rules to lie.
Here is a link to the rules.
You announce the card you are playing, then it is played. Checking for legitimacy is a separate act. Play proceeds as follows:
Play the chosen card face-up on the table (not on a pile).
Check whether it can be legally placed on an existing pile. Do so if possible, increasing that pile by one.
Otherwise, check whether it can be legally placed to start a new ...
This depends on which edition you are playing.
There are different editions published by different people, and the R & R Games rulebook answers this clearly:
Important: The player must clearly point to the cards which he is giving information about. (Thus saying "You have zero of something" is not allowed as you cannot point to anything.)
A lot of this will depend on your group and how you chose to look at the rules. Other than giving information about what is in another players hand there is nothing mentioned about players talking to each other. There is a note in an English translation of the rules that I found that does somewhat address this.
Hanabi based on communication - and non-...
The German version of the rulebook says its allowed
Der Farb-Hinweis, dass ein Spieler überhaupt keine Karten
einer bestimmten Farbe hat ist erlaubt!
Beispiel: „Du hast keine blauen Karten.“
roughly translates as
The color clue, one player having no cards
of a certain color at all, is allowed.
Example: "You don't have any blue cards."