If you expect a lot of background blur from wide angle shots, then I'd suggest you forget that idea.
If you want a lot of background blurring for one or two people shoulder-head shots, where it is most effective and most useful, then consider a 50mm f1.8 lens, which you should even find cheap used. It's a good value choice for that purpose and it's my focal length choice for my own Sony NEX F3 ( same mount as the QX1 ) and DSLRs for those type of shots. When not using a 50mm f1.8 I invariably use a general purpose zoom, similar to a 16-50 or 18-55.
But do not under-estimate what is possible with a kit lens ( a 16-50 would suit you, perhaps ). The larger sensor of the QX1 and A6000 are what is giving you some better background blur effects over a smaller sensor system.
I would say to you that while the QX1 look like a nice gadget on paper, in practice using an ILC implies changing lens, and changing lens means being able to practically use other lenses with your camera. The QX1 attached to a phone simply does not provide you with the needed handling.
An A6000 has a good viewfinder, is designed to be held using the viewfinder ( important for stable shooting and proper framing ) and also has features like focus peaking and good controls for photography. If you are going to do photography, get a decent tool for the job, not a gadget.
I would add that if you get into photography, a QX1 will seem like a dead-duck very quickly, whereas an A6000 ( or similar ) would prove much more useful.
So I'd at least consider getting a dedicated camera.