Haskell, 395 391 383 361 339 characters (optimized)
Still uses a generic state machine, rather than anything clever:
k="<>^v"
o(Just x)=x
s y(h:t)=case b of{[]->s(y+1)t;(c:_)->(c,length a,y)}where(a,b)=break(flip elem k)h
r a = f$s 0 a where f(c,x,y)=case i(a!!v!!u)"x /\\"["true",g k,g"v^><",g"^v<>"]of{Just r->r;_->"false"}where{i x y=lookup x.zip y;j=o.i c k;u=j[x-1,x+1,x,x];v=j[y,y,y-1,y+1];g t=f(j t,u,v)}
main=do{z<-getContents;putStrLn$r$lines z}
A readable version:
k="<>^v" -- "key" for direction
o(Just x)=x -- "only" handle successful search
s y(h:t)=case b of -- find "start" state
[]->s(y+1)t
(c:_)->(c,length a,y)
where (a,b)=break(flip elem k)h
r a = f$s 0 a where -- "run" the state machine (iterate with f)
f(c,x,y)=case i(a!!v!!u)"x /\\"["true",g k,g"v^><",g"^v<>"] of
Just r->r
_->"false"
where
i x y=lookup x.zip y -- "index" with x using y as key
j=o.i c k -- use c as index k as key; assume success
u=j[x-1,x+1,x,x] -- new x coord
v=j[y,y,y-1,y+1] -- new y coord
g t=f(j t,u,v) -- recurse; use t for new direction
main=do
z<-getContents
putStrLn$r$lines z
