starting up Chrome/Firefox/Any other browser in private mode using PortableApps

You can pass in what each browser uses in the command line yourself. For Firefox, that's the -private command line option. So you could start FirefoxPortable.exe -private directly on the command line. Or you could copy the FirefoxPortable.ini from Other\Source next to FirefoxPortable.exe and then add -private to the AdditionalCommandLine entry. Chrome works similarly with its own swicth.

You can also grab Private Browsing by PortableApps.com which you install alongside Firefox Portable. It will start private browsing with FlaskBlock, AdBlock Plus, and EasyPrivacy pre-installed. It's a bit outdated, so you'll need to update the extensions on first run. I'll push out a newer package of it this week.

Within App\AppInfo\Launcher\OperaPortable.ini, add CommandLineArguments='--private' to the [Launch] section...

[Launch]
ProgramExecutable=Opera\launcher.exe
CommandLineArguments='--private'
WaitForEXE1=installer.exe
...