Rav Moshe Feinstein writes in his reponsa in Even Haezer 4:60, that the basic reason the Torah (and/or Rabbanan) prohibits closeness with a single "girlfriend" (hugging, kissing, and even inappropriate talking), is because it will lead to inappropriate closeness and attachment; which will lead to sin.
Shmos 22:15 "If a man seduces a virgin who is not betrothed and lies with her, he shall provide her with a marriage contract as a wife."
The Torah shows us that in an extreme case of a man "wooing" a girl to the ultimate extent of inappropriate closeness, the "punishment" is that he marry her. In other words: If you cause an act of commitment then you must commit!
We know that a man, in the natural course of seeking a wife, must of course interact with potential women. However, he should not do anything that will cause too much personal closeness, unless he already intends to be serious about marrying her (even then there are balanced limits).
1. Is a man permitted to woo the maiden (/her parents)?
Yes. A man should try his best to a) Be himself b) Show he is of good character c) Engage in as much talk and activity with a girl as is needed to decide if he wants her as his wife. d) Be a good salesman and show her and convince her of who he really is and that he likes her etc. --- provided that he only displays affections equal to his seriousness about her.
If he does express extra displays of affection, and then rejects the girl, he will probably hurt her heart and have caused inappropriate feelings of closeness between them.
So, on a first date, he will get to know her. He will display courtesy, manners and his own personality. He will not express his undying love. :)
As the relationship escalates. He certainly should tell her what he is truly feeling; namely, that he wants to get serious/ get married to her.
Therefore, saying things and giving gifts that would usually be taken as serious bonding attempts of a personal nature, need to be toned down or eliminated until he has decided he wants her as his wife.
For instance, taking her out for ice cream is fine. Giving her a bracelet is inappropriate. (even if not for the sake of kiddushin with witnesses)
I heard in Yeshivah, that the following deal was made for all time between Adam and Eve. They discussed, that in the future, if humankind would go out on dates; who should pay for dinner? Eve remembered that she was told she would give birth in pain. Adam was told that he would work hard for a living. So, they concluded: If the woman has to go through childbirth, the least the guys can do is pay for dinner. :)
Therefore, buying ice cream for her is a normal standard by which a guy shows he is a "mentch" and has class enough to pick up the tab. (manners, not intimacy). However, buying her a personal gift, would create an attachment that may make her think they owe each other something prematurely.
Comments are similarly judged. He does not say: "You look beautiful in that dress". He does say: "You have good taste in how you match your outfit." He does not say: "I think you are very attractive." He does say: "You have great energy this evening."
That is the dividing line between what wooing is acceptable and what is not. Be normal when scouting a relationship without acting prematurely.
2. Is it preferable that he (or family) do it?
The Torah tells us: Bereishis 29:26 "And Laban said, "It is not done so in our place to give the younger one before the firstborn."
Bereishis 18:8 tells us that the angels ate food when they visited Abrahahm.
Rashi explains: "and they ate: They appeared to be eating. From here we learn that a person should not deviate from custom. (see Gemara Baba Metziah 86b)
The answer to the question of how exactly one should choose to act in shidduchim, depends on the local custom. You need to ask Rabbis, neighbors, and friends, about what is acceptable and proper behavior in your local society. There is no set of Halachos that cover all possible details of these situations. Rather you should know your local custom. Some places have the girl meet in her home. Some want to go out on a date. Some talk on the phone first etc. In some places flowers may be accepted, in others it is not acceptable. etc. In some places, the parents have a lot of involvement before the couple meets. In others, it is up to the couple etc. Everything is according to the Jewish custom of that community and family.
3. Or should he (and his family) just be himself and not do for her any special favors and let her want to marry him by herself?
He should just be himself, while being his best self. He should do her favors, without doing her "special" favors. However, just standing there like a fire hydrant while she "decides" to marry him "by herself", is ridiculous. What girl would respect such a person? Communication must take place both ways so they both want to marry each other.
Yes he should open doors for her and act with manners and chivalry. If he doesn't plan on continuing this practice, then he should change his mind and continue to do so! :)
What is over doing it?
Bereishis 38:23 "So Judah said, "Let her take [them] for herself, lest we become a laughingstock. Behold, I sent this kid, but you did not find her."
In general, be discreet in matters of one's relationship with a woman. Do what you have to do, but not in a way that the matter causes embarrassment to her or to you (or the family etc.) .
4. Should he also inform of his negative traits (which he is not Halochikly obligated to)? (Or should he avoid them to speed up his marriage?)
He should of course ask his Rav advice on what issues he should and shouldn't reveal (even if he is not obligated to reveal them).
The Ben Ish Chai (Niflaim Ma'asecha 71) tells a story of a man who was very desperate, and broke into a widow's home to steal money. He regretted the act, and left her valuables on the table. The next morning, she awoke and went to the Rav to find her a shidduch since she needed a husband to protect her! The Rav suggested the nice young man who happened to (unbeknownst to anyone) be the burglar himself! (The moral of the story was that if you are destined to have money, you will receive it without having to steal it, etc.)
Rav Yitzchak Zilberstein asks if the chassan should honestly reveal to the kallah that he was the burglar?
He answers that he does not need to reveal it. He proves this from the Urim V'Tumim (Urim 28:3). A Bes Din, does not accept testimony from a biased witness because they cannot be sure the witness isn't twisting the story. However, if the witness has a bias against the defendant that the court is not aware of, he can still testify as long as the witness himself knows he is telling the truth. He need not admit his bias to the court. The burglar knows he sincerely repented in the middle of the act. Therefore, he is allowed to become the widow's husband without telling her he was the burglar.
I hope this helps. :)