I upgraded from Magento 126.96.36.199 to 188.8.131.52, can you tell me if should I install also the SUPEE 7405 or it isn't necessary? How can I know if my site is vulnerable?
The SUPEE-7405 patch is already included in Magento 184.108.40.206. Magento has even released this version so that their latest downloadable version always contains all the necessary patches.
Also noted on the Magento download page (select "Release Archive"):
The patch addresses different security issues. Your site is no longer vulnerable to these issues. You can just trust on that.
To check if the patch was applied, check the file
You can also run your website through https://www.magereport.com/ which also includes a SUPEE-7405 check if you want some tangible proof.
No, you don't have to.
SUPEE-7405 is included in Magento 220.127.116.11