Does accidentally pasting password into browser URL field send it over the network?

In most browsers, yes. This happens because the browser sends the address bar text to a predictive service (commonly Google's) for search suggestions. Some browsers (eg, Firefox) ask before doing this (just the first time).

How dangerous is this?

Depends on where your browser is sending this suggestion text and how paranoid you are. Eg, Google should use:

https://www.google.com/complete/search?client=chrome&q=%s

And you note that this uses HTTPS, so no one else on the network can see your password. I can't imagine that any browser would be configured by default to use a suggestion service that doesn't use HTTPS, but if any did, you'd have to worry about your password thus being visible to anyone listening on your network.

But even with HTTPS, your password can certainly appear in server logs at Google and employees could potentially see it. That said, realistically, there's not a great deal of risk. Google's got far too high of a volume of requests to pay attention to any in particular, nor is there any way to know that any given query is a password. So it's very understandable that you might pass it off as so low risk that it's not worth changing your password (although it is a non-zero level of risk). I talk about Google here specifically, but the same applies to other major search engines, too.

In short yes it could have been sent to the domain shown in the browsers address bar and/or the search engine you have set up as the default in the browser to allow the quick search results.

You acted on the side of caution and reset your credentials which is something I would have suggested just to be safe.