I apologize if this sounds harsh, since that's not my intent, but if you're looking for "answers to puzzles" in your dealings with Japanese people, you're probably just going to end up more confused and in more awkward situations. As with any culture, sure, Japan has certain baseline values and attitudes, but the most important thing to keep in mind is that Japan is a country of almost 130 million people, and for each of those people, their personality is a result of their own thoughts, opinions, personal experiences, and even regional cultural influences which differ by specific part of Japan.
Because of that, any strategy like "When Japanese people say A, they really mean B," or "If you want to convey Idea X to a Japanese person, you have to present it in Format Y" is going to eventually get you into trouble. It's always best to think of a Japanese person as an individual person first, and Japanese second.
So, with that said:
uThe article suggested to ask three times, implying that the Japanese person would refuse twice, and they might agree on the 3rd time. Is this true?v
Sometimes. It depends on the person. But if you're asking if Japanese etiquette has an iron-clad rule of "always refuse a gift twice, and accept it on the third offer," then no, that's not true.
uIf somebody says they don't want something, do they really mean that or are they just "shy" and you would need to ask again or present the gift?v
Again, sometimes, but it depends on the person.
uMore later, I started noticing that a Japanese person might say, 'You don't have to [do X]', which might mean both 'You are not obliged to do X (but if you really want, go ahead)' and 'There is no need in doing X (so don't do it/don't go ahead).'v
Yes, some Japanese people sometimes do this. But then again, I grew up in the U.S., and some American people do this too. Pretty much everyone has a personal line that separates what they think of as a kind favor they'd be happen to receive, and something that they feel is too large a gift, even if it's being given willingly.
For example, you've thanked multiple people in this thread for their advice, and your messages of thanks have been accepted. But what if you offered to send me 20 dollars as a thank-you gift? Sure, I'd like to have an extra 20 dollars, but I don't want you to go to that trouble, even if you were to say it's no trouble at all.
On average, Japanese people draw that line of "I don't want you to go to any trouble for me" a little lower than other cultures, but again, it depends on the individual.
u I re-read the response of my Japanese lady friend. I'm afraid that I should have re-confirmed that she really didn't want me to send her anything... as from my current, better understanding, I think she was just polite in her reply. This makes me sad, because she really wanted that thing and I could not pass it on to her...v
There's no need to beat yourself up over this. You offered the gift, and she politely declined it. Even if she did, deep down inside, want it, she told you not to bother, and if she has any resentment or sadness over that result (i.e. she's upset that you didn't force her to accept the gift), that would be considered impolite even y Japanese standards.
Again, without knowing the specifics of your relationship with this person, it's hard to understand exactly what his thoughts were. If I had to guess, though, was there a long period of time between when you received the item and uWhen I had a chance to see him in person again, I asked him face-to-face how I could pay for the thingv?
This isn't a specifically Japanese thing, but some people feel uncomfortable receiving compensation for something they did a long time ago. For example, imagine that you and I are coworkers and go out to eat lunch. You forget to bring your wallet, so I pay for both of us. When we get back to the office, you ask how much your meal was and pay me back right away. No problems, no awkwardness.
On the other hand, imagine 10 years pass between when I pay for your meal and when we talk about how much your meal was. At that point, I might just be like "Eh, don't worry about it. It was a long time ago, so you don't need to pay me back."
It is odd that you specifically messaged your friend about paying him back and he didn't respond. Again, if I had to guess, maybe he felt that writing a message with the details felt too much like he was the one saying "give me the money," and would have been happier to discuss the matter in-person or over the phone. Again, though, that's not a specifically Japanese thing, so it sounds more like your friend is just an individual with some unique communication quirks.
Bonus gift-giving advice
It seems that you have a generous personality and enjoy giving gifts. However, while Japan is in many ways a gift-giving society, there's one aspect of Japanese etiquette that will probably be helpful for you to keep in mind.
In Japan, when someone gives you a gift, it's good manners to give them a thank-you gift in return. Sometimes you can anticipate this situation. For example, if you have a wedding reception in Japan, part of the planning is preparing thank-you gifts to give to your guests as they go home, since you know they'll have just given you wedding gifts.
But this applies even to smaller, unexpected gifts. For example, sometimes my wife's friend goes shopping, buys a shirt, doesn't wear it for several weeks, and then when she tries it on at home again, decides she doesn't like the fit as much as she thought she would. A lot of times she ends up giving the shirt to my wife, and then the next time my wife sees her friend, she'll bring her a thank-you gift. It's usually something small, like some cookies or chocolates or something, but still, it's something.
So if you're giving a gift to a Japanese friend, odds are they're going to feel like they should return the favor sometime down the line. On a basic level, that's nice and fun, but in practical terms, it can be sort of difficult, especially since the more trouble someone went to in giving you a gift, the nicer thank-you gift Japanese people feel like they should offer in return.
So while you might be totally willing to make something by hand and mail it to Japan for your friend, once she receives it, she's going to feel like she needs to do something just as thoughtful/involved for you, and that may not be something she's in a position to do. She might not know your tastes well enough to find something you'd like as much as a hand-crafted item, and she might not have the time to figure out what forms and postage she needs to mail a present to you.
And yes, sometimes this custom results in a pattern of Person A giving Person B a gift, Person B giving Person A a thank-you gift, and then Person A giving Person B a thank-you gift for the thank-you gift. Basically, simple "just for fun" gifts in Japan can often turn into extended shopping/mailing projects, and it's understandable that sometimes people would be reluctant to start the cycle, and will instead just say "Thanks for thinking of me, but that's OK, I don't need anything."