I can't help with an assembler recommendation, but Osh Park (https://oshpark.com) is what a lot of us use for low volume boards. I've used them for years with zero issues.
Most of the time, I also use oshpark.com to fabricate I have also used a few others like PCBs.io and seeedstudio.
Seeedstudio (https://www.seeedstudio.com/fusion.html) also does do assembly as well. Up on another forum I know a user uses them (or at least used to) to produce some boards. It was my impression that the costs to do so, can be greatly reduced if you use parts from their standard parts list.
I use "Easy EDA". I have had a number of boards fabricated from them.
They have an online design suite which includes a pretty good library of parts.
Support is also pretty good.
I believe they also do assembly too.
Prices so far have been pretty reasonable.
I just ordered from EasyEDA.com for the first time and got very nice PCBs at a great price. The order was for my company and subsequently we got charged a substantial import fee by the company that handles shipping and importation for us.
Do you guys have any issues with import frees when you order these boards?
if you are willing to spend a minimum of ca 120 euro's for 100 cm^2 PCB total PCB surface you can also use http://www.pcb-pool.com/ .
They do PCB and provide you steel stencil (free for 4-layer boards) and offer you assembly.
Their online configuration gives you the final price.