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1) Let $G$ be a Fréchet Lie group. Let $H$ be a closed subgroup. Is it always true that the centraliser of $H$ is a Fréchet subgroup of the lie group?

2) Is the closed subgroup theorem valid for Fréchet Lie groups?

I'm unaware of the the literature in this field and would appreciate if someone could point me to some compendium of results regarding Fréchet Lie groups.

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Concerning your second point, it is no longer true that closed subgroups are Lie subgroups (even in the Hilbert setting). The corresponding result in infinite dimensions needs additional assumptions concerning the interaction of the exponential map with the subgroup, cf. Theorem IV.3.3. in "Neeb: Towards a Lie theory for locally convex groups".

I'm not aware of any general results concerning a Lie subgroup structure on centralizer or normalizer subgroups.

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