Do all verbs in English have opposites?... Is there a similar situation for adjectives or adverbs?
Not all actions (or traits to describe with adverbs/adjectives) have opposites, so no. But, for all of those that do have opposites, yes, there's always a way to verbally indicate both. Even if there aren't truly independent words for them, like "arrive/leave" or "pick-up/put-down", we can always use oppositizing prefixes/suffixes like "dis" or "anti", even if the verb we're attaching them to doesn't use them very often and isn't found in dictionaries with them attached.
For example, not long ago, when I was talking elsewhere about a mysterious language's possible relationship with a known language family, I listed a few cases where the known languages in that family have two of the same or almost the same consonant in the same word, whereas the mysterious one, if it is indeed related, would seem to have just one, as if the fact that the consonants were repeated caused one to be dropped in this language. I called this phenomenon "derepetition". I don't think I or the people I was talking to had ever used or seen/heard that word before. I had no hesitation about constructing & using it, and nobody had any trouble understanding it. Similarly, in an online thread about the actions of certain characters in a TV show where one of them had betrayed a couple of others, I said that it looked to me as if he had then "unbetrayed" them later in the same episode. Again, without any prior experience with that word, everybody proceeded to use it for the rest of the thread, whether they agreed that the character had done it or not. (We even somehow got the actor who played that character, Adam Baldwin, to come on and use my word to say I was wrong about it, because his character had NOT unbetrayed them.)
Similarly, when a word or phrase contains a component with an opposite, you can replace that component's opposite to invent the whole thing's opposite, such as turning "overwhelming" into "underwhelming". For that matter, that example just reminded me of a time when I worked in natural resource management, when we were looking at two maps of adjacent areas: I said something about where the edges meet, someone asked me whether I meant there was an overlap, and my answer was "There's an underlap".
You even used the same method yourself in the original post, even if you might not have noticed it because some prefixes are used so much that we might tend to not think of them as prefixes:
activate/deactivate, implode/explode
How common & familiar such a word is just seems to depend on how common the need for it is and whether or not there's also another word that does the same job.
"Impede" has lost it's direct antonym "expede", but there are at least a dozen other unrelated words ready to fill that role, "facilitate" for example.
...including "expedite"... which lacks an "impedite".
