You will be able to use all that space under the arduino module (height permitting). Currently it has a big pink courtyard around it saying “stay out.” But you just need to tweak the footprint to have just the sip sockets and then the area is available. Kinda like this:
https://forum.kicad.info/t/advise-one-choosing-components/51516/6
You don’t need stackable sockets (just short-pin sockets), unless you want the long pins available to connect patch cables, logic analyzer…
I have used pcbway for bare boards, but not assembly. I have mostly moved to jlcpcb as prices are better and I have done assembly there with perfect results. You need to find their part numbers from their available inventory (which is huge). Parts cost is generally cheaper than mouser, though you have to pay assembly costs and dhl shipping. Still good pricing overall. Specialized parts (adc…) are not always available so I usually manually finish the boards, but having almost all of the passives and garden-variety parts assembled saves me oodles of time. You can get an immediate online quote, but you need all of the docs available first – gerbers (directly from kicad), and placement file and bom (in JLC’s format). I presume pcbway is similar.