Hi Mas - I was brought up in Joydens Wood and went to Dartford Tech - now Wilmington Grammar :-). Dartford was my old stomping ground quite a few years ago - LOL
From your description it sounds like you have one similar to this
These are very common and they can be quite challenging when it comes to fine temperature control. As you suggest the biggest problem is air control over the coals - some provide too much which leads to a very hot cooking chamber and others do not allow enough air to reach the charcoal resulting in them running too cool. It sounds like your problem is the latter. One of the smokers who came to the
UK smokers weekend in Billing Aquadrome last month brought one of these and we helped him to achieve a reasonable amount of temperature control without making physical modifications to the smoker itself.
Drilling holes in the bottom would certainly help with the air flow however this would not give you any fine control without also adding an adjustable vent control. If you can take a few photos of the inside of your smoker - water pan and charcoal bowl and any bottom air vents - we can help you more specifically. Sometimes it is simply a case of adding a mesh grate in your fire bowl to lift the coals up and let air get underneath them.
Yes, water in the water tray will affect the temperature that the cooking area reaches. The water tray itself (even when empty) acts as a heat baffle that protects the food from direct heat radiation from the coals. Just because it is there it does not mean that it has to have water in it. Some people use sand in place of water - or just leave it empty.
For low-and-slow cooking water in the bowl will help a smoker stabilise at or around 100-110 C (212-230 F), however if you are looking to achieve higher hot smoking/roasting temperatures of 160 C (320 F) then it will certainly make the higher temperatures more challenging to achieve.
Looking forward to seeing the photos :-)