When you update your 1GP, unless your 2GP is also going to be updated to make use of new features (object model additions, new APIs etc.) from the updated 1GP there is no actual need to update your 2GP's dependencies (or release a new version of your 2GP) at all. Dependencies effectively identify the minimum version that you depend on in this scenario since 1GP versioning is entirely linear.
On the other hand, if your 2GP is to be enhanced to use new features from your 1GP then you will need to first provide at a minimum a new beta release for your 1GP then update the 2GP to depend on that new version before you can update it to use the new features, then release a new 2GP version.
Yes, a 2GP can depend on a 1GP beta release (we have had to use this approach when providing an integration 2GP between our own 1GP and a third-party 1GP, since the third party would not give us access to a full release version). Obviously if that 1GP beta includes changes that are later abandoned and that the 2GP was changed to use you will find you have a "dead branch" in the 2GP version tree.