Hello
Lots of questions here... I'll try my best to address them all. Firstly, foot numbness or pain happens for a lot of different reasons, but in a nutshell, results from the force of the pedal underfoot causing excessive pressure or stress on the sole in a weird or unbalanced way. One of the nice characteristics of most cycling shoes is that they aim for stiffness in the sole -- a stiff enough sole, it's figured, more evenly directs pedaling force along the bottom of the foot and resists pain-causing stresses. Running shoes, by contrast, are pretty soft-soled, and there's no solid contact between your shoe and the pedal. It's likely putting a lot of funny strain on you.
That said, pain and numbess plague many riders, including those spending hundreds on high-tech shoes and pedals. One could guess that's because good pedal and shoe systems encourage harder riding, but that's only half of it. The only way to really prevent discomfort, in my opinion, is by finding the perfect (and fairly stiff) shoe -- it's all about fit.
Your next question gets pretty subjective -- people love their favorite pedal types like fisherman love a certain lure, or car enthusiasts worship certain makes. Ease of entry/release is a good starting point for comparisons, though let's be clear: you've already gotten some bum advice. Most mountain pedals are dual-sided, yes, but so are many road pedals. There's nothing stopping manufacturers from building all road pedals with two functioning sides, actually; it's just that with road stuff, an emphasis tends to be put on elegant designs with minimal weight.
One basic, relatively inexpensive format that's employed in many, many dual-sided pedals (both road, mountain, and general-purpose) is Shimano's SPD; you could walk into any bike shop and buy a pair of ultra-simple SPD pedals for very little. Anyone looking to help you save a buck will point you towards them. They're fine for many people, and make good general use or trail pedals, but I wouldn't recommend them to you -- mostly because you're concerned with foot pain, and inexpensive SPDs ain't too ergonomic. They're small, and they use a primitive, restrictive locking mechanism. Everyone's different, and they might be your thing, but I just can't see most SPD pedals as an ergonomic upgrade over sneakers and platform pedals.
I've opened a can of worms, though, because that leaves you to consider pedals that many folks wouldn't suggest to a beginner... not because they're hard to use, but because they're pricier, or simply because they're cool. Screw that. Figure out how much you'd like to pay, and get the rockin'est pedals you can. We can talk more about choices later; this answer is already too long.
Last answer. Getting in and out depends on the pedal system you choose. Nearly all feature an entry which is a variation on stepping down. Some, like the newer Shimano SPD-SL series and the Look models they're based on, are a bit like a ski binding -- you set your toe in, and pop the heel down (nearly all pedals of this style are single-sided road models, FYI). Others, like Speedplay (my personal favorite -- dual-sided and light), you simply step on.
For every model I can think of, exit is the same: you turn your heel, usually outward.
Ok, this is a big subject; I can write more and more and more about different features, pitfalls, benefits, and so on, but I'm friggin' sleepy. Hope this was a good start. Take care...
