Ok here it is the status of the things. I would love to have come comments from you about it.
- HP 8300 SFF 32 GB RAM i7 3770.
- Gigabyte GeForce GTX 1060 Mini ITX OC 6G video card arriving tomorrow. Sizes are H=37 L=169 W=131 mm. It should fit as it is made for MINI ITX.
- Special cable that allows to replace the crappy HP 240 watts PSU with a custom one ordered and arriving in 5 days.
http://www.ebay.co.uk/itm/112204268477. Many people used it successfully and got rid of the HP junk PSU.
- I am still looking for a PSU that I could buy. Any suggestion mates ?
With the special cable I have bought and a new PSU I should be more than fine. This video is made a British guy that bought my very same cable 
https://www.youtube.com/watch?v=0VCslFs6lW0