I for one don't really mind if a significant other wants me to take a traditional role (primary responsibility for bread winning, activity planning, decision making, and wallet opening) as long as she also assumes a traditional role (primary responsibility for shopping, cooking, cleaning, laundry, etc).
The problems come when there's a disconnect in expectations. The most common form of this in my experience manifests itself in what I like to call "American Princess Syndrome", where (some) young ladies in our post-feminist world seem to believe they're entitled to all the benefits of a traditional man (read: having their lives financially subsidized), while eschewing and scoffing at their own traditional responsibilities, for which they demand "equality" aka a shared burden. I think that's where the real danger of asymmetric contribution lies. You certainly don't want a woman who thinks you should have to "contribute more" (monetarily or otherwise) than she does. In my experience that speaks to a rather whorish mindset, i.e. "you need to pay for the privilege of having me", which is another tell of the American Princess.
Decide whether you want a girl who will cook and clean for you, or a girl who will split the bills. A girl who will happily do both (consistently) is probably few and far between--just as you might be unhappy bearing the full burden of your traditional role AND splitting her traditional responsibilities. Just make sure you don't end up with an American Princess, who won't do anything!
As far as the original questions--my favorite cheap dates have already been identified. For a first date, my top choices are going to a coffee shop in the afternoon, walking through a park, or both. The purpose of the first date, for both parties, is to see if there's even any chemistry--so why would either of you expect to spend a lot of money on something that might not even go anywhere? I will pay for one overpriced cafe drink for each of us on a first date, usually nothing more.
Second date is traditionally where I spend a little more, mostly so I can see how she behaves in public, and in the whole "traditional-date" situation (e.g., does she offer to pick up part of the bill, etc). Bowling, cheap restaurants, that sort of thing.
Invitations to either person's house for dinner might work, but personally I would save that for a second or third date, or immediately after a successful coffee/park date, or frankly on whichever date you expect sex to occur. From my perspective, it can be a little awkward (if not downright uncomfortable) going to a stranger's house or having a stranger over, even ignoring all the preconceived notions of where such a date is "supposed" to go.
After the first date(s), it's a lot easier and cheaper, if there's chemistry and mutual interests. My girlfriend and I got bicycles together. Her parents have horses that we visit and ride occasionally. We go to parks and hike. We go to lakes and swim. We've gone on camping and museum trips. Being introverts, though, we mostly stay at home and watch movies, play video games, cook together, and, uh, "entertain" each other. We've been together over a year and I've taken her to a bar exactly once (and we didn't pay for drinks), and out to restaurants (counting fast food!) maybe three or four times in the entire courtship. Of course, if you're dating an extraverted shopaholic or American Princess, none of this will apply.
Which brings me to question 3, meeting ERE-compatible people. Again, "ERE-compatible" will be different for everyone. What is it you really want?
I have to say I disagree with the bias against online dating. It's all too easy to dismiss online dating as something that's "only for freaks", but I can't help but find that amusing in a thread that's essentially about how difficult it is for us to fit in with mainstream expectations. How do you think 99% of the world looks at us? If you want to find another freak, go where the freaks are!

Seriously though, the world has changed. The stigma against online dating is fading. I doubt you'll meet someone compatible with ERE by cruising singles bars--other cheap introverts with busy lives who actually want relationships don't cruise singles bars anymore, they make accounts on POF or OKCupid. Again, think about what you want.
I may be a little biased myself though, because I've never done traditional dating of the "meet a stranger in public/through a friend, get a number, arrange a date" variety. I've done internet dating the last few years and before that, in school, all my relationships started as friendships first.
(I catch a lot of flack whenever these dating/marriage threads come up because I tend to speak my mind and call things like I see them without much heed for political correctness. So in conclusion, I'll simply say that everything I've stated is my own opinion and experience, which I am in no way presenting as objective or universal. These are just my general observations. Sorry in advance if anyone takes offense.)