What three suggestions does Brutus reject in Julius Caesar and why did he reject them?

Brutus disagrees with Cassius about swearing an oath, including Cicero, and killing Antony. 

Relative to Cassius and the other men who start the conspiracy, Brutus is a relative latecomer.  Cassius convinces Brutus to join by arguing Caesar must be stopped because he is a tyrant who will abuse his power.  Cassius tells Brutus that his name will lend legitimacy to their enterprise.  Brutus is not content to be a figurehead, however; he actually wants to lead. 

Nobility is very important to Brutus.  He wants to make sure the conspirators are seen as tyrant-killers and loved by the people.  For this reason, he wants to do everything in the proper way.  Cassius tries to advise Brutus, but time and time again Brutus does not take the Cassius’s advice.  He does things his own way. 

The fact that Brutus will not follow Cassius and has his own noble and idealized ideas about things is evident from the very beginning.  All of the conspirators meet at Brutus’s house in the middle of the night.  Cassius wants the conspirators to swear an oath to their cause, but Brutus countermands him immediately. 

CASSIUS

And let us swear our resolution.

BRUTUS

No, not an oath: if not the face of men,
The sufferance of our souls, the time's abuse—
If these be motives weak, break off betimes,
And every man hence to his idle bed;
So let high-sighted tyranny range on,
Till each man drop by lottery (Act II, Scene 1). 

As far as Brutus is concerned, the men should be loyal to the cause, not an oath.  Brutus wants to believe everyone's motivations are for the same noble reasons as his, which is actually quite naïve on his part.  The men have a variety of motives for wanting Caesar dead.  Some are patriots, but some are just greedy.

The next objection involves Cicero, the renowned orator.  Cicero was known to oppose Caesar, but he was also very famous.  Cassius suggests involving him, but Casca, Cimber, and Cinna immediately naysay the idea.  Brutus sides with them. 

BRUTUS

O, name him not: let us not break with him;
For he will never follow any thing
That other men begin.

CASSIUS

Then leave him out (Act II, Scene 1).

When Decius suggests they might want to kill other men, too, Cassius jumps on the suggestion.  He wants to kill Antony because he is Caesar’s right hand man and might be dangerous.  Once again, Brutus is completely against the idea.  He wants the killing of Caesar to be a noble act.

Our course will seem too bloody, Caius Cassius,
To cut the head off and then hack the limbs,
Like wrath in death and envy afterwards;
For Antony is but a limb of Caesar:
Let us be sacrificers, but not butchers, Caius.
We all stand up against the spirit of Caesar;
And in the spirit of men there is no blood (Act II, Scene 1).

Once again, Brutus is concerned with image.  He is more worried about the way the public will perceive their actions than the logistics of the actual assassination.  Rather than listen to Cassius’s counsel, he goes his own way.  In the end, Brutus's insistence on doing things his way will haunt him, as Antony is indeed dangerous.