Should I top up pickling cabbage solution?

Are you making vinegar pickles or fermented pickles? Sauerkraut is usually a fermented product, which uses salt, not vinegar. So, I am a little unclear.

In either case, I'm doubtful that the cabbage absorbed anything, but your vegetables should always be submerged. Usually that is accomplished by weighing them down. If they are fully compressed you can certainly top off your container with a brine solution. The reason for this is that the cabbage above the solution will likely develop mold.

The refrigerator is not a typical place for fermentation to happen. Pickling usually takes place at slightly warmer temperatures (though, if you are just making vinegar pickles, which are not fermented, the fridge is the right place). I keep mine in a cold basement room, for example. Refrigeration will dramatically slow the fermentation process.

Generally yes you should top up the solution, anything exposed to air even in the fridge will start to spoil. Many pickling jars have an insert and/or weight that you can add at the top to make sure the food remains submerged.