Yes and yes. I've seen pics of him playing both Teles and assorted Gibsons.
As far as "Layla" is concerned, Clapton played his beloved Strat "Blackie" on the entire album (I believe) while Duane used his '58 or '59 Les Paul Standard. Again, I've seen photos of Duane playing a Strat and Clapton often uses Gibsons (335s, Les Pauls, SGs and even Explorers) both live and in the studio.
It is really impossible to say with definate accuracy who was responsible for each solo on "Layla." Even Clapton can't recall (heroin, cocaine and alcohol were endemic in the studio during the recording sessions).
As a rule of thumb, Duane likely played most of the slide work and likely used his Les Paul whereas Clapton likely played most of the non-slide solos and likely used his Stratocaster. (I say "likely" as Clapton is an accomplished slide-player in his own right and Duane can solo well himself.) Both contributed to the rhythm parts.
I'm currently looking into the history of "Layla" on the net and will gladly let you know what I find.
Thanks -- I'd appreciate that. "Layla" is somewhere in my top 20 favorite albums.
MD