Yes, the FBI will investigate the current allegations about Trump because the American people will make sure it does. We, the people, have a voice, and we will tell the FBI to investigate since it is information we must know. We cannot hide from Trump, we must face him head on.
I could be wrong, but I think the FBI has to investigate him. Whether they are able to investigate him properly and whether they make their findings public is a different matter though. As the days go on it seems more and more likely that Trump is going to be the President for the next four years and they are going to be scandal-filled years.
Yes, the FBI will investigate the allegations about Trump, because it is their job to keep up on things like that. They need to investigate the allegations to make sure that Trump and the rest of the country is safe. They're also interested in the gossip of what Trump is up to because it's interesting.
From the way Comey responded during questioning yesterday, it sounds as though the FBI is already investigating these claims. He refused to confirm whether there is an investigation, stating that it is policy not to confirm so during an active investigation. Basically, I take that as his way of saying yes, they are looking into it.
The FBI is part of the executive branch so Trump is going to be their boss. The only ones who can investigate, or order an investigation of, the president is congress which is under republican control.
While I don't doubt that they will look into the allegations, either to disprove them or verify other actors, Trump will not be a focus of the investigation nor will they implicate Trump without being directly ordered to investigate him by Congress.