When Fred first met his German hosts, he shook hands firmly, greeted everyone in German, and even remembered to bow the head slightly as is the German custom.
Q1 : What is the function of 'as' ? Conjunction? Pseudo-relative pronoun?
Q2 : I'd like to know the sentence structure after 'as' (= is the German custom) Is this inversion?
It could be an adverb since it means "like" as an adverbial preposition. You can substitute "the way" for "as," and that is adverbial also.
HoonyQ1 : What is the function of 'as' ? Conjunction? Pseudo-relative pronoun?
I'd say your interpretation "pseudo-relative pronoun" is pretty good because "as" acts like "which".
... to bow the head slightly, which is the German custom.
HoonyQ2 : I'd like to know the sentence structure after 'as' (= is the German custom) Is this inversion?
Under that interpretation there is no inversion because "which"="as" is the subject, and of course "is" is the verb.
Merriam-Webster has a definition of "as" as pronoun, thus:
as (pronoun): a fact that
Ex: is a foreigner, as is evident from his accent
Personally, I think a definition of simply "which" would have said the same thing.
CJ