Questions about "Speed Dial" screen on Nexus' Phones app
Is there any way to configure it to go straight to the dialpad when launched?


No, at the moment, there isn't. You will need to first press the Phone icon, which will load up the stared contacts screen, then press the dialer icon at the bottom center of the screen to get to the dial-pad

Or at least the normal phonebook?


Contacts app does this. You can place it on the homescreen for quick access. 

It almost seems to pick contacts to display on it at semi-random, and as far as I can tell, there's no way to manually override its decisions about whom to show on the page.


If you star a bunch of contacts, they show up at the top of this window. You may rearrange them by pressing and holding on the contact tile. What follows the starred contacts is the set of contacts whom you frequently contact. 

If you want to clear the frequent contacts, press the 3 dot overflow menu on the top right and tap on "clear frequents"

A simple way to avoid going to the dialer is to directly search for the contact name from the persistent google search bar at the top of the screen. This will in turn show the contact you're looking for along with  the telephone number details from where you may place the call.

Another option would be to open the phone app, then search or directly type in the entire number(if the contact isn't stored). If the contact is indeed stored, their name will directly show up. If not, you'll see something like  "Call 9xxxxxxx".

If you fancy handsfree calling, it is as simple as saying - "Ok Google, dial 9xxxxxx" or "Ok Google, call Mr.X on his cellphone"

One other option is to try out some other custom dialer apps! There are plenty on the play store.