"If she immerses in a Mikveh and a court of three observant Orthodox Jews finds her Giur to be successful, then she is a Jew, right?"
Right, though you'd want to make sure the beis din is widely accepted. See the list of RCA batei din (http://www.judaismconversion.org/batei.din.html), or the list of batei din accepted by the Israeli Rabbanut (http://sephardicjewishguide.weebly.com/conversion.html).
"But are they expected to totally turn someone away if they begin conversion as a result of being in a relationship, because it is 'for the wrong reasons'?"
No. If the person genuinely wants to convert, is willing to become completely observant for life (even if the couple splits up, for example), and the spouse is willing to become observant enough to avoid getting in the way of the spouse's conversion, then most Orthodox batei din will allow conversion. It is a fairly common situation that a Jew and non-Jew are dating and the non-Jew converts Orthodox before marrying. See the book Becoming a Jew by R' Lamm, which notes how common this is.
See the following policy statement by the RCA, which manages a network of widely-accepted batei din:
"Where marriage to a particular Jewish partner is a major incentive to a prospective conversion, there is an increased possibility that the geirus may come with less than the complete commitment necessary for a conversion that would be in keeping with the standards we are trying to set for the regional Batei Din. Nonetheless, experience also shows that such a motivation can result in converts of the highest caliber. Conversion for the sake of marriage therefore requires the Beit Din to constantly reevaluate if the candidate and future partner are likely to subscribe to the requisite beliefs and practices. The Beit Din must be convinced that if the potential spouse were to disappear from the candidate’s life, his or her commitment to the Jewish faith and people would not waver. These factors inevitably prolong the process and make examination of the prospective convert more intense. Indeed, should the couple mention a proposed wedding date as a deadline or goal, the Beit Din should respond that the process will take significantly longer than that."
"What is the most successful way to actually get such a Gentile girl to convert?"
I don't understand what you're asking with this question. Do you meant if that a Jew is dating a non-Jewish woman, what is a successful strategy for persuading her to convert? If so, this is a problematic question. Most sincere converts convert out of a strong personal desire to convert, not because someone wanted them to. However, there may be situations in which a non-Jewish woman exposed to Orthodox Judaism in a non-forceful and gradual way may come to decide she wants to convert. As noted below, however, this is unlikely to succeed and one should not expect it to take place.
Perhaps by the question you mean the following: if the non-Jewish woman wants to convert, what is the most successful strategy for completing the conversion? In this case, the answer would simply be to attempt to start the gerus process and don't take no for an answer (that is, persist in your interest even if rabbis try to discourage you or don't answer your calls, etc.). How to go about starting and progressing in the conversion process is explained at length and very practically in The Gerus Guide by R' Aryeh Moshen.
"If I were to date gentile girls, and casually tell them about Judaism, and only marry one who becomes genuinely interested of her own volition (placing me in the company of e.g. Steven Spielberg, Jared Kushner and Sasha Baron Cohen) what are the consequences of this in the afterlife, etc. according to rabbinical Judaism?"
It is wrong for a Jewish man to date non-Jewish women. So as far as the afterlife is concerned, I would think you would be punished for that (if you do not do adequate teshuvah in your lifetime, at least). It is important to note that the vast majority of non-Jewish women you date will not be drawn strongly enough to Orthodox Judaism to be able to successfully convert. It is an extremely risky and dangerous strategy to date non-Jewish girls with the hope that you can nudge the one you end up falling in love with and wanting to marry into converting. In short, it is extremely unlikely to succeed.
"The effect on soul of the convert of not following all the commandments post-conversion vs remaining a relatively righteous Noahide"
If the convert did not truly accept all the mitzvos, then the conversion was not valid. But if the conversion is valid and the person is not completely observant afterward, the person would be punished for their aveiros as would any born Jew. Note that when a convert is unobservant after a conversion, the conversion is sometimes considered to have been invalid.