DropDown and keyboard

I was expecting to be able to tab to the dropdown and use the keyboard to start writing in the search box (if the dropdown allows filtering). However, now I need two mouse clicks, one to open the dropdown and another one to position myself in the search box. Not very form friendly.

Also, when you tab to a dropdown you don't have any focus indicator.

Is there any fix or workaround available? These are much needed for basic a11y(accessibility) testing. Appreciate if anyone can respond to this request.