It always helps visualizing what the words really mean or where they come from. And if you do that, it is relatively easy to see that "erinnern" describes the process of "going into oneself" — in order to get something which is stored there, but not directly relating to the memories themselves. "Erinnern" is about handling yourself, not your memories, thus the reflexive pronoun. "Remember", on the other hand, does not combine "re" + "membrum", as I first thought, thus describing how we put parts together again, but derives from "re" and "memor" (not "reminiscor", apparently), the latter basically meaning "keeping something from being forgotten". So, "to remember" indicates the direct handling of thing of the past, thus the accusative.
With "to feel", it's not so black and white as it appears in your question. There is an intransitive ("How do you feel?") and a transitive version of "to feel", in which case you also say "I feel myself ... " in English. And that easily explains why you need a reflexive pronoun with "fühlen" — it is a transitive verb.
So, it always helps to dig a little deeper into the matter as opposed to just saying, it is like that in my mother tongue, so it should be the same in the other language. Unfortunately, the other way round, German native speakers (and French, Italian, etc., by the way) are confronted with the problem of "false friends" where English terms and expressions are just literally translated without thinking of the proper German equivalent.