Can I charge my phone with many different charger speed types? I have a nexus 6p, and various different chargers which I bought for the phone. One is a slow USB charger (comes with the box). 

Another is a type c to type c cable connected to the AnkerCore 20100+ battery pack (this charges quick charge too but I charge that with a USB mini so the current is different). 

I also have a charging dock and sometimes use my computer to charge it, along with a wall charger with the highest speed. 

Does this effect the battery in any way and does it effect the time before I have to charge it again?
All the chargers should work, to a varying degree... if they are QC compatible, the phone may charge faster than "normal", if not, the phone will attempt to draw current up to the maximum capacity of the charger or the phones "normal" limit. 

Does this effect the battery in any way and does it effect the time before I have to charge it again?

Yes, of course it does... Different sources will vary the recharge time, one charger may charge the device to near full in a significantly shorter time than another device, causing you to have to recharge sooner or more often. Charging very fast, such as Quick Charge, can have a negative impact on the long term life of the battery itself.