NO...THAT IS NOT NORMAL...
NORMALLY...The hull will collect water from condensation....But from your pic...And assuming that the amount of water that collected at the bow was from just ONE NIGHT...I suspect that water is leaking from around the hatch/hatches and/or places where screws and bolts penetrate the hull.
What I noticed shortly after I purchased my PA-14 was that when it rained alot (+2") or after I took my PA out in less than desirable conditions (+10mph winds w/ +2 ft. waves...Water to the degree of your pics was showing up inside my hull. So...I purchased some silicone goop and felt around the inside of the hull, wherever I could reach, and covered the screws and bolts. Then I shortened the bungees that hold down the hatches to make a tighter fit, and also added weatherstripping at each hatch cover. That fixed the problem of excessive leaking from screws, and also from loose hatches. I still get leaks...But only when I take the PA out in conditions where no small craft should be...
And...YES...If your battery is not centered, or you have alot of water sloshing around inside your hull, that can throw off the center of gravity, and balance of your PA which will make your PA less responsive or feel out of balance. But just having a paddle hanging over the side of your PA is not enough of a weight difference to affect the balance of your PA. I suggest your removing the battery and any extra gear (weight)...Then take your PA out and see if it still feels lopsided and/or out of balance. If removing the battery and all other gear (weight) fixes the problem, then you will know that you need to balance out the weight...Its like this...If you load your PA, or any other watercraft, and everything is loaded on one side of the watercraft, the watercraft will be out of balance (it will lean to the side where all the weight is)...You need to make sure that you load your PA evenly. Try to center your battery. If you can't, and have to load it on one side of the PA, try to counter the weight by adding something of the same or close weight on the other side to keep the balance equal...BUT...IF when your PA is empty, other than you sitting in it, and it still leans...Something is wrong and you should contact your Hobie Dealer...
I hope that I helped answer your questions