Recently I've found that there are many questions asked by new users (most of them with 1 point reputations) and most of them ignore the basics of StackExchange mechanics. This kind of accounts ask a question (and sometimes many followup questions more) and then leave without marking the question as solved. I don't care about the points but when I'm looking for answers myself I tend to trust more in answers marked as correct than in open questions. Is there any procedure in place in order to purge that kind of questions?
The situation is a real mix, which makes it tough to have a blanket solution. Whether or not a question is clear, whether or not an answer is accepted, and whether or not the OP tries to expand the question in follow-up comments should be considered as three completely separate things. I'll talk about these in reverse order.
Expanding the question
Sometimes, follow-on questions indicate that the original question didn't accurately describe what the OP was trying to ask. Sometimes people view the site like a forum, where the question is the "price of admission" and then they can expand it in comments to ask every other question they ever had. In the latter case, we need to educate the OP on how the site works and tell them to ask a new question, but the original question may be well answered.
Failure to accept an answer
Failure to accept an answer is a separate problem that occurs widely, irrespective of question expansion or question clarity.
Acceptance only signifies what the OP found most helpful. New users sometimes accept answers for the wrong reasons, so it may not even mean that. It doesn't necessarily mean that the accepted answer is the best one for someone else.
Accepting an answer can indicate the intended meaning of an ambiguous question, but that isn't the purpose or meaning of acceptance, and the issues should be viewed separately.
The site is a knowledge base with the purpose of sharing solutions with other people experiencing similar problems. If the OP solves their problem, that's great. But if the knowledge can't be easily used by others, it doesn't contribute value to the site. So if a question is useless to anyone but the OP, it should probably be closed even if the OP has accepted an answer.
Accepting an answer can shed light on the intended meaning of the question, but the answer isn't part of the question. The question should stand on its own to be useful to other readers. It should not be allowed to remain unclear just because an answer was accepted. The question should be edited to clarify it based on the accepted answer.
Clear question, lots of activity, no accepted answer
On the other hand, a question can be clear and have one or more answers, but the answers may be the problem. Answers aren't necessarily good answers, or they may be filled with correct information but not address what was asked. There can also be "group think", where the first answerer misinterprets the question or posts a solution that biases the thinking of later answerers. So lack of an accepted answer may be warranted, even though there was a lot of activity on the thread.
Not really, they might be valid questions. It could be that the answers provided have not solved the issue, in which case forcing an accepted answer on them would be misleading to others.
If there are people answering questions but those posts are asking for more information or seem more like comments you can flag them as "Not An Answer" or "Low Quality".
They then appear in the review queue for you and your peers to look over.