1. A fire basket would be preferable but probably more helpful in longer burn times rather than better temperature controls. Now if you just have a pile of just sitting on the bottom of the firebox, yeah you should definitely get some type of elevated, ventilated box in there to assist you. Expanded metal is pretty cheap and you can get 24x24" sheets at your big box stores. It is located with the other small sheet metal pieces. Some tin snips and bolts, washers, and nuts can build you a box with no metalworking skills.
2. I do not have a BGE, but that is a pretty big investment and you are probably drastically reducing your cooking area when compared to your offset smoker. I would probably recommend a less pricey model to try first before jumping into a $1500 item.
3. I feel the same way about electric smokers. I recommend them for beginners who don't really want to get into the bbq a whole lot.
Some extra advice to help save you time and effort. You mentioned your smoker leaks a little bit, well they all pretty much do that but if it is relatively airtight in the firebox region there are several automatic temperature controllers out there that will handle keeping the fire going for you. I have two models of the BBQ Guru brand. Essentially a fan that controls the air going into the fire based on a temperature probe that goes into your cooking chamber. They are very effective. Another device that can help you from running back and forth is a nice wireless thermometer like the Maverick 732 that will monitor meat and pit temperatures and transmit them to a receiver that you carry around with you. The Maverick is probably the most popular but there other brandnames too.
If you like the pit you have now, I would stick to it and try to improve on it.
XL Big Green Egg, Weber kettles, WSM 22.5" and Pellet Pro adapter