I'm not aware of any way to save a general search within the SE system. You can of course save a browser page and reload that, so that it has your favourite query.
However, one thing you can do is set your favourite tags.
Your favourite tags show up in the right hand sidebar under the "hot meta posts" section --
(those are ones I set a while ago but forgot about so they're still there. I don't really pay enough attention to that.)
-- you can change them right there in that section by clicking the edit link, you don't have to do it from your profile each time. They will be highlighted on the page when you're looking at the list of questions:
The exact form of highlighting varies from SE site to SE site. For us, there's a bluish background.
(If you click on a tag in your favourites list you'll get a search on that tag.)
You can also exclude some tags in that same mechanism. Their posts don't disappear but they're greyed out and easy to ignore. (Unfortunately -- from my point of view -- the excluding overrides the favouriting; I'd usually like that to work the other way around)
I use this facility on SO (to get questions related to R) and math.SE to get questions related to stats, probability and a few other topics (on CV I sometimes set favourites to highlight some tags for a while but I haven't ever excluded any except to see how it works)
I think that some top users here on CV have a saved search which allow them to find the unanswered questions with some specific tags,
Usually if I want to find unanswered questions with a specific tag I just type something like
[tag-name] answers:0 ... how long does it take, maybe two seconds?
most of the questions are answered by specific users, [...]
I don't think people are continuously scrolling down the list of unanswered questions to find those pertaining to certain tags
Even without a search it doesn't take long to scan a page full of questions. If you just answer what you can and you know how to answer the questions that will fall under specific tags, your answers will naturally be under those tags even if you never search for a tag.
I usually don't go looking for old posts, but if an old post is bumped for some reason (e.g. someone edits or answers or Community bumps it because it doesn't have an upvoted or accepted answer) then it will show up on the front page ... and then people who load the front page shortly after that will see it and might answer it. Often I don't even realize it's an old question for some time.
PS if I want to search for [tag1] or [tag2], the search is simply [tag1] or [tag2], right?
Looks like it, yes. e.g. see “Wildcard” and “or” in search
(which also discusses wildcards)
and Possible search engine bug when searching for multiple tags (>2)