To answer your questions, yes Beethoven performed his own compositions (as well as that of his contemporaries).

It seems reductive to call his renditions of his own music "rote," as in any two performances of the same piece, he would play the same notes in the same order and with the same rhythms, but pretty much everything else (tempo, dynamics, tone color, etc) was improvised to one degree or another.

Yes, he did sometimes improvise outside of the above stated realms, when he was performing on keyboard. These instances were mostly cadenzas (which were basically extended, drum-solo-like decorations prior to major structural points in pieces). At this point in music history, the cadenza was very often left up to the performer to improvise based on the themes of the piece. Prior to Mozart, cadenzas were completely improvised. Mozart only composed a small handful of cadenzas for his sonatas and concerti. Beethoven composed quite a few of them. And after Beethoven, it became very very rare for cadenzas to be improvised.

The crowd for Beethoven concerts varied by venue. He often performed his solo sonatas in salons for small crowds, but his concertos and symphonies required more space and thus were played in larger halls.

At this time, this was the pop music of the day, so any impressions you may have of tuxedo-clad aristocracy sitting quietly and absorbing the genius of Beethoven should be jettisoned. It wasn't quite a bar-type crowd where people tried to speak loudly over the house band, but it wasn't that far removed. People listened, but also had conversations and played games of chess and other light distractions from the music. It was much more relaxed than today's concert halls.

Beethoven worked with other musicians very often. These were sometimes duet partners (as in a violin sonata with piano), chamber groups (trios and quartets). Sometimes he performed a concerto on the piano with a complete orchestra as accompaniment. And of course his symphonies required full orchestras (and a full choir in the case of his 9th symphony), which he would sometimes conduct with a baton.

A fascinating resource if you're really interested in this kind of thing is First Nights by Thomas Forest Kelly. It goes through the first performances of 5 pieces throughout music history:

*Montiverdi's Orfeo from 1607

*Handel's Messiah from 1742

*Beethoven's 9th symphony from 1824

*Berlioz's Symphonie Fantastique from 1830, and

*Stravinsky's Rite of Spring from 1913

The book delves deeply into the concert experience of the times: who attended, class distinctions, how many musicians performed, what they were paid, why there was a riot at the Rite of Spring, etc.

Edit: formatting

The first thing to remember about Beethoven is that he was nearly stone deaf by the time he was 40, so public performances were impossible. There is a famous story about him conducting the premiere of his Ninth Symphony, and being so deaf that he kept conducting past the end. One of the singers had to stop him and direct him to face the audience who were offering a nearly ecstatic ovation. In reality, Beethoven was probably not actually conducting, but he seems to have been standing at the conductor's side offering assistance with tempos. The part about not hearing the audience reaction seems to be true by all accounts.

The next most famous account of a Beethoven performance is 1808 (he was 38 and nearly completely deaf), a monster four hour concert in which he offered the premieres of the Fifth and Sixth Symphonies, the Mass in C, as well as the Fourth Piano Concerto. There were also some other works, such as the Choral Fantasy, a work for orchestra and chorus that was a precursor to the Ninth Symphony. Someone else conducted the orchestra, while Beethoven performed on piano in the Concerto and the Piano Fantasy, and also for an intriguing portion of the program that would include "improvisations" by the composer. Beethoven arranged the entire concert himself, renting the hall and hiring the orchestra. Despite it's now legendary status as the greatest concert in history, the actual performance received mediocre reviews. It was inadequately rehearsed, with some parts finished that day (reportedly the ink was still wet), and the Choral Fantasy had to be restarted when the orchestra completely fell apart while sight reading it. It was also a freezing December evening, and the hall was extremely cold, and we can only imagine how this must have affected the tuning of gut string instruments and the flexibility of the musicians' fingers. In any case, the program is still performed occasionally, despite it's marathon length.

This turned out to be Beethoven's last public performance, but it tells us much about him as a performer and composer. Since he was paying for the hall and the orchestra himself, he wasn't about to perform and promote the music of another composer. This concert was meant to show off the best works of his extremely fruitful middle period, in hopes of selling more music and getting more commissions. It also shows that he was a first rate pianist, to be able to handle the virtuoso demands of the Fourth Concerto and the Piano Fantasy, as well as the public tightrope stunt of improvisation, for which he was legendary. The concert also demonstrated that Beethoven's place as one of the great composers was already established. The concert was well-attended and the music was praised, although the length of the performance in the cold hall was criticized. One critic mentioned that "one can easily have too much of a good thing." His deafness kept him from having more of a performance legacy, but in fact that may have been a blessing in disguise. Beethoven was such a surly, disagreeable character that the Viennese orchestral musicians were openly hostile toward him. Future concerts would only have been more difficult to arrange.