I would take a gravity reading and taste it!
it looks like it fermented well, did it come out of the top?
if you had the white cap on it then that is definitely OK. CO2 is heavier than air, so would have protected the beer, and your white cap would have stopped bigger bogies from getting in.
but all in the taste, right?
The yeast will be using the sugar to make alcohol and carbon dioxide. I'm not sure where you read that it will get sweeter, but as far as I know the more sugar you add the more the yeast will consume. The best way to sweeten it is to add a sweetener such as Stevia. You can only use sugar/honey/maple syrup if your yeast has died.
Be careful adding sugar ...
So far my brewing is all from kits, I don't use air locks, just 5gal buckets with lids, not sealed all round - so pressure doesn't build up. Then I prime and bottle the flat beer and it gets its fizz from there.
As the initial fermentation has co2 sealing the surface of the brew the free space above won't matter, then if you bottle condition it should all ...
I know this sounds too easy, but I had half of a batch do this. I was using two different types of bottles (8oz. and 16 oz.). It ends up that the smaller bottles had thin, cheap, rubber gaskets, and there wasn't a good seal. No seal = no pressure buildup = no carbonation.
I used almost the exact recipe/process the OP used. I noticed the problem when I ...