Regarding the question
Is it OK to send an abstract to an editor asking if the corresponding article could be considered for publication in the journal?
(my rephrasing).
My answer is: Yes, it is ok, but this is unsusual and I am not surprised that you did not get a response.
To give some advice: You can do a similar thing, but, instead, send the whole paper and not only an abstract to an editor and ask if this article is within the scope of the journal. I did this, and I got a reasonable response from the editor very quickly (a few days - the paper got rejected finally, but on the ground of well found reviews). I guess that just sending an abstract would not have gotten any response in my case, too. This is because it is virtually impossible for an editor to judge the content or contribution from an abstract without the paper - well, you can describe some contribution, but an abstract is much too short to argue in a proper way.
Regarding your further questions:
Am I doing something terribly wrong here?
Not terribly wrong, but unusual.
Can anybody give me advice as to how I can "handle the situation" from now on?
My advice: Write your article with a specific audience and journal in mind, then send it to the (associate) editor of said journal that fits best and ask if the paper would fit the scope of the journal. If he or she says no, rewrite the article for another journal and repeat (or find a journal for which you think that article does not need rewriting). If he or she says yes, but do take it for granted that the paper is accepted. The chance of acceptance is not changed by asking in advance.
Do such e-mails effect my chances of publishing there negatively (or outright stop them)?
No, it would not get you blacklisted for the journal (I am not sure if such a thing actually exists) and your chances to publish an article are most likely not changed (unless you send multiple requests for an opinion of on unfinished work an really annoy any (associate) editor.