Are anime OP/ED theme songs original?
The vast majority of the time (easily upwards of 95% of the time, by my estimate), OPs/EDs are produced specifically for the anime they're used in. Reuse of existing songs is rare. 

There are counterexamples, though, which by their very rarity prove the rule. A few select examples:


Erased reused Asian Kung Fu Generation's 2004 song "Re:Re".
JoJo's Bizarre Adventure (2012 version) reused Yes's 1968 song "Roundabout".
Speed Grapher reused Duran Duran's 1981 song "Girls on Film". (Thanks, @ConMan.)
One Week Friends reused Sukima Switch's 2004 song "Kanade", though in this case, it was re-recorded with Amamiya Sora (voice actor of the character Fujimiya Kaori) as the vocalist.
And, of course, Neon Genesis Evangelion uses a great many different recordings and mixes of the classic "Fly Me to the Moon" as its ending theme.