As the question states, What does Alfred do in Batman? Alfred at first glance seems just to be an ordinary butler of the Wayne Manor who gives good advice to Bruce, but is it just that? Just a butler?
Is Alfred only a keeper of the Mansion as well as the bat cave?
It depends on continuity.
Originally, he served as a comic relief. In most early tales, he made bungling attempts to be a detective on a par with the young masters.(wiki)
Pre-crisis, Alfred Thaddeus Crane Pennyworth was a trusted support staff. NOT on "could not manage being Batman without" level, however.
Post-crisis (after 1983), his role was much bigger. After the death of Bruce's parents, Alfred was appointed his legal guardian and became a father figure, raising Bruce to become the man that became Batman.
Aside from serving "merely" as a butler, Alfred also serves as :
Helper when science and engineering is concerned. He maintains equipment such as BatComputer. Especially in Pre-Nolan movies, he pretty much seems to assume Lucius Fox's role.
An expert on first aid and medicine. Without him, Batman would risk detection much more due to having to go to the hospital.
At times as field helper. He is frequently portrayed as a capable combatant. In many versions, he's ex-special-forces.
He is also serving as archetypal "wise old man", giving Bruce advice, helping him keep his moral compass, serving as a psychologist, sounding board and the only person Bruce can trust.
Your last question "Could Batman manage being Batman without Alfred?" doesn't really have an objective answer. May be he could, may be not. It most certainly would have been more difficult for him.
As far as a literary/narrative device, Alfred functions to keep Batman/Bruce connected to humanity. When he's holed up and isolated in the Bat Cave, here comes Alfred, providing mentorship, advice (usually not asked for, but he always brings up ladies of interest such as Selena or Vicki Vale, etc---I'm mainly thinking B:TAS here), help, dialog, etc.
Without Alfred, Batman would be isolated. FYI Robin(s) perform a similar role when done best.
Alfred is primarily father figure and coverstory provider. However, Alfred is the one who teaches the Bat-family the art of disguise(the main thing) and acts as dispatch pre/post-Oracle.
There have been instances where has acted as security for the mansion and has worn a Batsuit to act in his stead.
