The latest version of Geeklog v2.2.0 will upgrade fine from Geeklog 1.4 but as always be sure to back up your files and database first incase you have to put everything back as before.
The GLMenu has not been upgraded for a long time so you would have to uninstall this plugin first.
There is another menu plugin you can use though which can be found here:
Geeklog Plugins that Support v2.2.0
The only real issue I see is your theme. There has been many theme changes since Geeklog v1.4 so your current theme will not work. Your best bet is to use one of the current themes of Geeklog (like the denim theme) and then change it to look like your older theme.
All the current themes of Geeklog are reponsive and therefore mobile friendly.
One of the Geeklog Core Developers.