Same as above, you can mix and match but I’d use the specific country’s passport if travelling there.
I personally renew them as soon as they expire, but I travel a lot for work so having one document out of date complicates things a lot when I need to leave passport X at an embassy, passport Y at another, and use my ID card for Europe travel. So I’d just keep them valid because you never know when you might need them.
When travelling to a third country eg Japan, use the same passport to enter and exit, especially if it’s been stamped. Even if it hasn’t been, you don’t want your US immigration records to show that you never exited the country and ‘overstayed’ your visa for example.
I have two EU passports and my biggest mistake so far has been entering the UK on the passport that isn’t linked to my settled status, by mistake. Cue a thorough interrogation as to what the purpose and length of my visit are, and do I have a job back home? 
I have a go-to ‘default’ passport (Spanish) so I mostly use that one. Later this summer I’m actually planning on entering the other country I have a passport for, with the Spanish one. They will not care, but all countries involved are EU.