OK:
1. What do you think are among Beethoven's greatest achievements? Most "influential" works?
His symphonies probably. Many Romantic composers found it difficult to escape the "shadow of Beethoven." Some, like Brahms for instance, felt this influence rather deeply.
2. What are your favorite works by Beethoven? Least favorite?
The piano sonatas, symphonies and string quartets. I can't get more specific than that. Least favorite? I really haven't heard anything that I don't like by him.
3. Do you feel that Beethoven's personal challenges (deafness; nephew issues; etc) considerably heightened the emotional content of his music?
Perhaps, I am sure that his upbringing also had this effect as well.
4. Any favorite recordings? Of Symphonies? Piano Sonatas? Concertos? Chamber? Overtures?
Symphonies: Szell, HvK '62, Barenboim
Piano Sonatas: Annie Fischer, Gulda (Amadeo), Gilels, Backhaus, Schnabel, Kovacevich, Kempff (m), Hungerford
Concertos: 1- Richter/Munch, 2-Pollini/Jochum, 3-Annie Fischer/Fricsay, Serkin/Bernstein, 4-Arrau/Haitink, 5-Serkin/Bernstein, Pollini/Boehm
Chamber: SQ- Vegh, Italiano
Overtures: Szell/Cleveland