Can I select the cover photo of Year, Month?

It is not possible - Photos is picking the key photo automatically, based on the objects in the photo.

I also wish we could change the key photos, as Photos is primarily picking photos with happy, smiling people or cute animals, at least in my library, so the photos picked by photos are not typical for the year or month at all and pretty useless.

You may want to send a feature request to Apple, using the feedback form: Feedback - Photos - Apple

Really, the best thing may be to make your own albums. You can use a Smart Album:

You can put these smart albums in year Folders, and sort them alphabetically in each folder. If you like, you can put all the year folders together inside another folder, sorted alphabetically, to keep them separate from other albums you may have.

In your own albums and folders, you can choose key images, sort, and even rearrange them as you wish.