Now I understand that the job requirements are pretty high for the best, most well-paid jobs.
The best jobs just don't get advertised in Poland - they go to people who are already known by others.
I lived in China for 3 years and have worked for 1 year in corporate training, 6 months teaching adults in a language training centre and several months teaching children.
Won't count for anything in Poland, I'm afraid - Asian experience is seen as irrelevant here, not least because the mentality is completely different.
I have a 7-day intensive TEFL certificate, however I don't have a bachelor's degree as I quit university after 1 year.
This will count heavily against you here. When the school secretary will have a Masters degree and every Polish teacher too, you need to have the CELTA as the bare minimum to make up for your lack of degree. Without CELTA and a degree, there aren't many doors open to you I'm afraid.
With this level of qualifications and experience what kind of work would I be able to find?
In the beginning, you'll be looking at Callan school work at best. You can expect it to pay no more than 2200zl/month on average (taking into account all the holidays, cancellations, etc) - which -
Also, I'm ideally hoping to live in a reasonable-sized city, not some boring small town.
Will be nearly impossible. Most cities in Poland (except, perhaps, Bydgoszcz and Zielona Gora) are full of native speakers already - and they are qualified native speakers too. Any city with something happening in it will already have many native speakers there - but "boring small towns" may be willing to hire you without degree/CELTA.
If you really want to come to Poland, try and get a job anywhere you can. Do that for a year, and doors will open elsewhere. But you've got no hope in hell of getting anything reasonable in any city that can be called attractive.