Be careful deleting your own questions, especially if they have answers (whether you agree with or like those answers or not), because this can result in the system automatically banning you from asking questions, if a significantly high proportion of your questions end up this way.
If a question is:
- Topical for the SE site you post it on;
- Of good quality (well-written, clear, and answerable);
- Not primarily opinion-based;
...then it is likely that the question will not be put on hold or closed. You can not put your own questions on hold. Only diamond moderators and those with the Close Vote privilege are able to put questions on hold.
If you decide to abandon your question and you do not want to delete it, you can just leave it on the SE site you posted, and not take any action. Leaving the fate of the question up to the community in this way is generally fine.
If the question is a poor fit for the site or suffers from unfixable problems, it will likely be eventually closed and then later deleted. If the question is a good fit for the site, it will very likely receive some answers that are at least plausible, if not correct, unless the question is extremely niche -- niche questions may not get many answers or comments, but the existence of the question and the dearth of answers will at least allow people to find the question on Google and connect with other people who have the same problem in the comments, at the very least. So, having questions on the site with no "easy" answers is fine, if the questions don't have any quality problems that would make them a candidate for closure.
Note that the function of "accepting" an answer serves an extremely minor role in the grand scheme of things on StackExchange, so you should not feel personally obligated to come back and accept an answer. The answer voting system is far more important for determining how valuable an answer is, than the singular opinion of the person who first asked the question. It is a common occurrence on SE for the original poster to find one answer helpful, and then 2 years later someone else posts a much more correct, more up-to-date answer that ends up receiving twice as many upvotes as the originally accepted answer. This is just how the site works. The green checkmark is not important (at all, in my opinion).
If you find the answer to your question on a non-SE site later and you want to self-answer your question, you can do that, and it's perfectly allowed within the site rules. You're allowed to answer your own questions on StackExchange, and as long as you keep the quality bar high, we all appreciate your contribution to the community! :)
You can cross-post something from SE to a non-SE site or vice versa, that's fine, but please don't cross-post from one SE site to another; if your question is more suitable for another SE site, it can be migrated by flagging the question and requesting its migration.