Assuming one works on a website and decides to be polite and friendly by telling visitors it is still under construction, does it impact its ranking negatively? Does Google and others take this notice into account one way or the other for ranking?
Once your site is live, do everything humanly (or inhumanly) possible to keep it up and serving content.
If you take a whole site down and replace it with an "under construction" landing page, expect to see your SEO suffer as inbounds are redirected to the content-less landing page. Google will take note of this over time and either drop you like a rock (best case) or de-index you (worst case and more likely).
The best practice is to do your development and testing on some form of staging server, whether that is a simple local server environment or an actual replication of the live environment only accessible to your team. Once your new stuff is ready, push it to the live server. Doing this will bypass the need for any sort of "under construction" or "coming soon" notices.
Now, if this is a brand new site and you are publishing a "Coming Soon because we're not done yet" type of page, then that's a different story. Google will see a content-less "Under Construction" or "Coming Soon" single page site as incomplete and probably will not index it as is. But you should not be putting up a content-less Coming Soon page. This is an opportunity for you to pre-seed your desired market with some actual, if generic, content and let Google chew on that while you get the rest of it up to speed. See my answer on a similar question for some suggestions of what to post while you build.
It is likely. An under construction notice would not indicate to me that a site is not a quality site. Either it is brand new and has no content yet, or it is being worked on and the webmaster isn't happy with it yet. If the webmaster isn't happy with it, maybe I don't want to go there. All websites could use some improvement. If the site is worth visiting, then forego the notice that there could be more coming if and when you get around to it.
Google certainly detects similar situations. Their blog posts about what has changed in search include changes to their algorithm that detects parked domains. They also pay attention to server statuses such as "503 Service Unavailable".