My theme version is 1.5.5 and wordpress version is 3.7.1. I intend to upgrade wordpress to 3.9. Is the 1.5.5 version compatible to wordpress 3.9?
Yes - usually there should not be any problems with the theme and WordPress 3.9.
Where can I check that?
You could setup a test installation and test WP 3.9 first before updating your production site.
Also, I've made a lot of customizations to the theme, so will upgrading the theme cause issues?
If you did not use a child theme (or the custom CSS box) for your customizations and have modified any core theme files, then your changes will be lost after a theme update.
Similarly will there be trouble if I upgrade to a the responsive/premium verison?
The premium version of the theme has much more features and options and also a different codebase. So in the end it is actually more a new theme which you will have to install on your site.
Have a great weekend!