If I may, sir:
I have no experience in dating or initiating courtship with women. However, I do have a somewhat formidible bit of personal experience with social anxiety and experiencing the same sensations you've captured here (if not similar circumstances).
I would like to invite you to take a step back and avoid defensiveness for a moment, if you would:
- You recount women (whom you yourself may or may not have been interested in) becoming, for lack of a more formal word, flirtatious with you (physical contact, eye contact, smiling, presumably some verbal interaction), even though they weren't interested in you. But how do you know they weren't interested? The way you describe the situation would suggest otherwise. Did they (either directly or indirectly) let you know this somehow, or is it something you yourself have just assumed? If it is just an assumption, or even what you would define as an educated hypothesis, how did you arrive at it? Explain the answer to this question to yourself in detail (and if you want, share it).
- The single mother who was expressing interest in you - how did you arrive at the conclusion that she simply knows that you yourself aren't good at contributing to setting-up a date? Are you certain she was really reading into your personal story, or could it be that she was merely listening to it? Again, ask yourself how it is that you arrived at the conclusion she is likely just trying to play with you. Breakdown and explain that conclusion to yourself.
In my own personal experience, I've found that after simply taking a step back and breaking down my conclusions and assumptions into their components they lose their plausibility and (consequently) tangibility. Your own mileage may vary, of course.
As far as your frustration and anger is concerned, I know where you're coming from in terms of having to live with it. I myself deflate and defeat it when it really crops-up by considering why it is I'm so mad. Typically, it's emotion I've built-up entirely on my own with a few powerful, but unfounded, inferences and speculations. While there are definitely manipulative and dangerous persons in the world, they're the exception rather than the rule; most people are not out to intentionally hurt or enrage you.
Try to avoid words like 'pathetic' when dealing with yourself - it's vague, unuseful and likely inaccurate. Use terms that are more precise (even if they may be negative); for example, you sound rather angry and depressed. What, specifically, is conjuring these feelings? Explore the answer, break it down. If these feelings aren't something you feel is a positive influence, what could you do alleviate them?
I apologize that I cannot lend a helping hand when it comes to approaching and dating women, I just hope my advice is useful to some degree as regards gradually overcoming your feelings of bitterness and hopelessness. Perhaps you can use this as a stepping stone to reaching your goal.