I am running version 45 of opera broswer on Linux Mint and do not want to install different versions on my machine
When you use the https/http settings to open in another browser, you're basically just using it as an email client. The email client requires a lot of the browser stuff anyway. All the standalone Opera Mail is anyway is the exact same Opera 12 browser + email suite with just some of the browser UI tweaked/hidden. So, I wouldn't worry about installing Opera 12 and using it just for email while using Opera 45 for browsing.
You can double-check though that it won't interfere. What profile folder is Opera 45 using in your home directory? If it's not ".opera", then you don't have to worry at all. If it is ".opera", you'd launch Opera 12 with the:
-pd "/home/your_username/.operamail"or
-pd "~/.operamail"command-line switch. See http://www.opera.com/docs/switches/#unix for more info.
As in, you can download the tar.gz for Opera 12 instead of installing the package. That way, you can extract Opera 12 to where you want and then run it (with the included script modified for a different profile directory) with what profile folder you want. (I forget if Opera 12 used a bash script to launch or just a shortcut to the binary.)