In act I, Cyrano directly challenges three men. Who are they and why does Cyrano conflict with each?

Cyrano directly challenged De Valvert to a duel after De Valvert insulted him. Cyrano dueled with De Valvert after he attacked his character by stating that he was a buffoon. This was after De Valvert made an unpleasant comment about Cyrano’s nose. The two gentlemen engaged in a fencing duel while Cyrano recited a ballad as they fought.

CYRANO:

[with grimaces of pain] It must be moved—it's getting stiff! This is what happens when it's been unused for too long! Aie!

DE VALVERT:

What is the matter with you?

CYRANO:

The cramp! I have a cramp in my sword!

DE VALVERT:

[drawing his own sword] So be it!

A Bore came up to Cyrano admonishing him about stopping Montfleury from performing.  The Bore stated that Montfleury was protected by the Duke of Candal. Cyrano did not seem bothered about it, instead, he asserted that his sword was enough to protect him against anyone who challenged him. Cyrano challenged the Bore after he hesitated to leave at his request. He asked the Bore to tell him why he stared at his nose. The Bore was unable to explain himself and was forced to run away from Cyrano.

CYRANO:
Get out now! Go!
BORE:
But I—
CYRANO:
Go! Or tell me why you stare at my nose!
THE BORE:
[petrified] I—

CYRANO:

[walking straight up to him] Well, what's so strange about it?

Cyrano forced Montfleury off the stage and challenged him to return. Cyrano believed that Montfleury was a bad actor. He also had another reason but decided to keep it a secret. 

MONTFLEURY:

I—

CYRANO:

[leaves his chair, and settles himself in the middle of the circle which has formed] I will clap my hands three times, you full moon! On the third clap, I want to see you eclipse yourself! ...

CYRANO:

[delighted, sits back in his chair, arms crossed] Come back if you dare!....

A YOUNG MAN:

[to CYRANO] But, Sir, why do you hate Montfleury so much?

CYRANO:

[graciously, still seated] Young man, I have two reasons—either will suffice. First, he is a terrible actor. He heaves up his lines as though they were buckets of water drawn clumsily from a well, when instead, they should soar from his lips like the lightest of birds. The second reason …well, that's my secret.