The answer to Question 2 is "yes". Take the relatively free group with law $x^4=1$ and infinite set of generators. That group is locally finite (Sanov, I. N.
Solution of Burnside's problem for exponent 4.
Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 10, (1940). 166–170.). 
It is not inside the group of finitary permutations (every permutation of order 4 is a product of 4-cycles and 2-cycles) and no non-identity element has roots of order 4.