For having such poor navy (no shipwright, dry dock, cannon galleon and bracer), it seems like Cumans cannot stand a chance on water maps.
If I have to pick a random civ as requested by my host and I get Cumans on the water map (eg archipelago, islands etc), is there a way to at least survive on said maps? Or I am downright out of luck?
Either you land them and knock them out like that, or you lose.
You are as good as most when it comes to ships through the Castle Age. In many games the naval war can be won by the end of Castle Age (Even Feudal). Tragically, you do not have an economic bonus that would aid in this endeavor as you would need to be aggressive. Landings are another path that is not mutually exclusive.
The Cumans stand a chance but may be on a timer depending upon the dynamics of your game.
Landing seems to be the best option, you could get some very fast xbows or knights on early castle thanks to the cheaper buildings.