instantaneous composition?

Your question covers a lot of territory.

So I'll break it down:

<<<Is composing spontaneous?>>>

In regard to Mozart, yes.

In regard to Beethoven, no.

Before Mozart put pen to paper, he had the entire composition in mind.

He could even write out the orchestral parts without writing the score first.

Not so with Beethoven.

He had to work and rework.

In his book, The Joy of Music, Leonard Bernstein devotes a chapter to Beethoven manuscripts in which notes are scribbled out and rewritten.

<<<composing at the piano>>>

Composers who work at the piano are often frowned upon.

A good composer could imagine how his music will sound without trying it out on the keyboard.

It is ironic that Beethoven spent his earlier years as a keyboard junkie,

but later went deaf, which rendered it impossible for him to be a keyboard junkie.

Mozart did much of his composing while riding in a carriage.

We have mixed data about Puccini.

He wrote the entire last act of La Boheme at a table where his friends were playing cards.

However, he was apparently a keyboard junkie a few years later, at the time that he was composing Madama Butterfly.

At that time, the first motorcars came out.

He was fascinated. He bought a motorcar and ran it as fast as possible down the country road.

He had an accident, so he had to sit all day with his leg propped up.

This delayed his work on Madama Butterfly, because he couldn't sit at the piano.

The moral of this story is: Don't become a keyboard junkie.

<<<studying music theory>>>

The Nineteenth Century composer Cesar Cui never studied music theory.

He had to sit at the piano and fish.

It took him ten years to compose an opera which a well-trained composer could have written in a fraction of that time.

I sometimes wonder how quickly it would pay off if he had taken that time studying music theory.

You may or may not have heard of Cesar Cui.

He is recognized for a short violin piece entitled Orientale, but that's about it.

It is said that Irving Berlin couldn't even read music.

Rather, he dictated his songs to someone else.

I read this story in a magazine ad for wannabe songwriters, so weigh it very carefully.

Different composers had different methods and processes, for those in the more distant past we may never know for certain. Many Bach manuscripts appear to be first drafts, almost improvisations, and many pieces of his we have no original. Beethoven is different, in nearly every draft of his you'll annotation and remarks about revisions, everything was meticulously planned and thought out.

While many composers use the piano, there are many who do not. Stravinsky like to use the piano, but Brahms, Beethoven, and Schoenberg rarely used it as a compositional aid.

Theory comes after the fact and made to explain how and why music functions the way it does. Part of the reason why many of the greats have been enshrined with their status is that they were pivotal in stylistic and technical changes. Then more theorists come along to explain what happened.

There are two main goals to music theory, 1) is to describe how music functions 2) to provide a guide to musicians to better understand the music they perform.