One of Daniel Barenboim's more unnerving habits, witnessed during his electrifying Beethoven-Boulez cycle at the Proms this week, is to stop conducting mid bar, usually when the orchestra is going full throttle and the tempo is breakneck. Typically he prods, pokes, throws punches at his players – gestures only, that is – before gathering them in like a bundle of loose lagging, casting his arm in a big arc and leaving them to it until the next cue.

It is a form of musical freewheeling which only the most experienced can risk. Generally it pays off: many of these musicians have worked together since Barenboim and Edward Said founded the West-Eastern Divan Orchestra (WEDO) in 1999, its aim to unite young Arab and Israeli players in the intensity of music making. The youngest, already a veteran of four years, is 15. He shares a desk with one of the concertmasters (lead violinists) of the Berlin Philharmonic, who takes a busman's holiday to be part of the Divan. The oldest is in his 40s. Is there an upper age limit?

"About a week or 10 days after death," quipped Barenboim at a press briefing on Thursday, when he said that while the Arab spring has made it easier for some countries, notably Egypt, to send musicians, others have drawn back completely, not least Syria. In 2014 a Divan Academy to train new players with talent but inadequate training will open in Berlin. The main orchestra is established musically and ethically yet its political credentials remain prey to shifting tendencies: Ramallah, once possible, is currently off the schedule. A concert in East Jerusalem this week has just been cancelled. For reasons of care, not carelessness, players' names are omitted from the Proms programme.

Barenboim first ingested Beethoven's symphonies decades ago and directed from memory with ease and freedom. No doubt he could do it with his feet if required. Part of his allure, in addition to his super-sized musical intelligence, is his apparent lack of self-doubt, which reveals itself as a dangerous compound of passion, focus and bloody-minded insouciance. Who else would slot a 45-minute work by Pierre Boulez, Derive 2, in between Beethoven's first and second symphonies – and what other broadcaster in the world, public or private, but the BBC would entertain the idea, as Barenboim retorted on Thursday, and he should know.

Boulez's chamber piece, full of snatches of melody and flutters and ripples of sound, seethes with detail, lush with vibraphone, harp and piano, though the whole remains elusive. It acted, surprisingly, as a bridge between the light, maverick textures of Beethoven's First and the more radical ambitions of the Second. Chronology of composition has nothing to do with it. Aural illumination is what counts.

All the other Boulez works were short in comparison. In Dialogue de l'ombre double (1982-5) the clarinet soloist (Jussef Eisa) converses with pre-recorded sounds wondrously manipulated by Ircam engineers through six loudspeakers. Beguiled audience members looked up as if hoping to see nimble and ghostly instrumentalists flying around the Albert Hall rafters.

Michael Barenboim, 26, who leads WEDO was the brilliant violin soloist in Anthèmes 2 (1997). He has been in the orchestra since he was 14 and now sits at the maestro's left elbow. "He's my dad but he treats me like any other musician," he has said in interviews. You look in vain for a special handshake or smile: he may be speaking he truth.

Two other chamber works, Mémoriale and Messagesquisse had the mysterious feel of private music written for a planetary king and his courtiers in an alternative universe.

One regret was that Boulez (b1925) could not attend owing to surgery. Barenboim and Roger Wright, director of the BBC Proms, rang him each night to report back. Even if some left the hall to escape this experience as if fearing fire, 6,000 others listened as hard as they knew how. Ever the realist, Boulez, according to Wright, amused himself with the calculation that this single performance constituted a bigger audience for Derive than the total up to that point.

WEDO's Beethoven project has been under way for three years. That dedication showed. Inevitably there are peaks and dips. Barenboim favoured broad tempi and a beauteous, vibrato-rich sound, suited to the scale of the Albert Hall, maybe, but it can sound dated to ears accustomed to the acidulous revelations of period-instrument performance – a practice for which Barenboim, regarding it as narrowminded and fogeyish, has never had much enthusiasm.

In the Third, Eroica, the skilful woodwind and brass sections were given luxurious free rein. The "middle period" Fourth (1806) had exactly the right muscularity and drive. Surprises were not always welcome, as in the over ponderous Fifth, when the piccolo player stood up for the prominent solo near the end. "If Beethoven wanted a piccolo concerto he'd have damn well written one," growled one of the UK's leading composers afterwards, among the musical elite which turned out each night for this Proms epic.

The Seventh had urgency and drive but remained earthbound while the Eighth, beloved of the composer, had grace, poise and buoyancy, an airy prelude to the shattering Ninth (on Friday, too late for this week's column. Catch it, and a BBC2 documentary Barenboim on Beethoven: Nine Symphonies That Changed the World on BBC iPlayer). Beethoven, that star-crossed genius who never abandoned hope, was honoured for his most important qualities: humanity and music.

Two Cadogan Hall Proms deserve long mentions but must make do with short. Harpsichordist Mahan Esfahani's arrangement of The Art of Fugue, premiered by Esfanahi and members of the Academy of Ancient Music, made Bach's counterpoint glisten so brightly you could imagine – faint hope – you could comprehend its intricate workings. Julian Philips's Sorowfull Songes, expressive settings of Sir Thomas Wyatt, had its world premiere at Monday's lunchtime Prom sung by Tenebrae, alongside Steve Martland's fruity Street Songs, all part of the London Pride mood of this 2012 Proms season.

It set the tone for another proud London institution, Opera Holland Park. On the day it reached the final stages of its 8,000-mile journey, the Olympic torch passed the theatre to the sound of Nessun dorma. Then Verdi's final opera, Falstaff, updated to the 1940s with knitting, vicars and tea urns, took to the stage, with joyous playing from the City of London Sinfonia – brass, especially – and an excellent chorus and cast led by Olafur Sigurdarson. As it ended, in a riot of red, white and blue bunting, it was as if the UK's musical Olympiad – and how can one resist a topical cliché in a brief tropical heatwave? – had gone for gold.