Yes. Keep in mind that voelen is a "toevallig wederkerend werkwoord" (no idea how that is called in English, but basically it means that it sometimes is a reflexive verb, sometimes it isn't (when there is a direct object)).
- Ik voel me goed = I feel good (reflexive)
- Ik voel mijn teen = I feel my toe (there is a direct object, so not reflexive)
The translation is grammatically incorrect. When speaking about a state of being (which the sentence obviously is since it's a reflexive), we use adjectives (good, bad, etc). "Feeling well" or "feeling badly" would imply that you are physically touching something else and your fingers' nerve endings are doing a good/bad/whatever job at feeling. This is a very common English error even among native speakers. But the correct version is at least as common, so I think the translation should reflect that.
Bottom line: "action verbs" take adverbs and "state-of-being verbs" take adjectives.
Scroll down to the bit where it says adjective. As well is recognised also as an adjective, the grammar here is correct; and this is, besides, a common phrase, so from a descriptivist point of view would be fine whatever.
Jospeh2, I understand what you are saying. However, the English word "well" is tricky. It can be the adverbial form of "good", but it can also be an adjective in its own right, as in, for example, "I am well", meaning I am in good health.
So here DL is in fact following your rule, and using the adjective "well" with a state-of-being verb.