It is possible to use https://apps.openerp.com/apps/modules/8.0/tg_pos_packs/ for example. It allows for the creation of bundles in the POS on the fly. You may also want to check other modules available in the App store for this purpose.

I did test the tg_pos_packs module for the restaurant use case, it works perfectly for example for pizzas with different toppings, as a "dish configurator" (such as changing side dishes and defining the meat doneness) etc.

You may also ask for Serpent's (http://www.serpentcs.com/) restaurant POS module which handles this issue using product properties.

No, Odoo POS won't allow you to work with variants. Although there are workarounds for certain scenarios like the one that you mention. 

You can have a main category that shows all your products and this is where your hamburger will go. Than you can have an Additionals category where you will put all your additionals like cheese and bacon. And last a Remove category where you can put your informational products like no pickles, no onions, etc... 

Another option would be to enable the notes in the POS and put the information there. 

And last but not least... Custom development to get exactly what you want. 

You have better use the POS from Vend or magento as well. Vend software is compatible with most of POS device. However, some Magento POS generates API code to other device are also convenient as well, for example Magento ipad POS: http://www.magestore.com/retailer-pos.html/