I usually play Civ 5 on Prince difficulty level with Raging Barbarians checkbox ticked. And I always ecnounter situation when AI players have zero gold and negative gold per turn however even with these problems they fight and conquer other civ cities. Shouldn't they have they army deserted, building destroyed or something like this? Or can I build more and more untis decreasing my GPT but having gold stay at zero level?
When your gold hits 0 and you're still in the negatives, the first thing that happens is your Science production is tapped to make up the difference.
Eventually, units will start to disband, but this happens over time. Thus, if they've only been running a debt for a short period they may still be able to effectively war.
Since capturing enemy cities usually results in negative things happening in the short term, it's probably not the best strategy for digging themselves out of debt.