If a Pokémon trainer loses a gym badge can they get it replaced? In Pokémon: Indigo League, if a trainer earns a gym badge but then loses it during their travels and/or gets it stolen, do they have the ability to get it replaced without having to battle the gym leader again? 

Throughout the series Ash keeps his badges pinned to the inner part of his jacket, but he's always flaunting them. Would it be possible for someone to just steal all his badges and skip the process of battling gyms, or is there some kind of registry that will verify the trainer's progress?
for the first question
In the first season ep75 "Bad To The Bone!" a guy named Soto has lost all of his badges to Team Rocket saying "All our struggles and sacrifices were for nothing", meaning he can't just go and claim his badges were stolen, but in the 5th season ep45 "Why Wynaut!" after Ash's badges were stolen Clair the gym leader tells him "If you report that your badges were stolen I'm sure they'll be reissued in time", coming from a gym leader it must be true, so the situation in "Bad to the Bone" is different since it's may not be in time, given that in Ash journey he lost his badges three months before the league started and in Soto situation it's a matter of a day or two.

for the second question
In the episode "Bad to the Bone!" after Team Rocket had stolen the badges from Soto, Jessie saying "I've finally got my very own set of badges, now I can enter the Pokemon League Competition and prove ... " and the whole Team Rocket plot to steal the badges was to enter the Pokemon League, saying it's enough to enter.  Although in ep77 "Round One Begin!" Ash is been asked by a clerk in the Pokemon Village an identification to register the Pokemon League and Ash gives her his Pokedex it's been clear since ep2 that the Pokedex serves as an ID too, so if a person will give an ID with his badges it will be enough although he may steal them as proved from "Bad to the Bone" episode.