After completing the sugya of Yayin Nesech which contains many details, it is recommended to go through the Chochmas Adam klal 75-77 where he brings all the cases of the sugya and many of them that are discussed in Tosafos down l'maseh (see here).
Here are the basic rules:
1. A goy can make wine assur by: a. touching it with his hand or something else he is holding. b. shaking an open bottle of wine that has a narrow spout. c. ko'ach - causing the wine to move such as pouring it into a cup.
2. Category "a" and "b" are prohibited to even derive benefit but category "c" is only an issur to drink.
3. The ko'ach of a goy (category c) with intent is prohibited to drink, but without intent is permitted to even drink.
4. Rashi holds that goyim nowadays are not truly idolaters and therefore have the status of a child who doesn't comprehend the service of avoda zara and therefore the wine he touches is only forbidden to drink but permitted to benefit from. The Rama rules that we can rely on this opinion in a case of loss. Therefore, in a case of loss, whenever the gemara will say it is forbidden to benefit, it is permitted to benefit but forbidden to drink; whenever the gemara will say forbidden to drink, it is even permitted to drink.
Q and A of some practical questions:
1. What happens if one leaves a bottle of non-mevushal wine in their refrigerator and they have a gentile cleaning lady? A goy cannot do anything to make the wine forbidden so long as it is closed (doesn't have to be sealed) because even "shaking" would not be a concern unless it is an open bottle. Therefore, if a Jew is in the house or can pop in so that the goy is scared to open the bottle, the wine is completely mutar even if the goy moved the bottle around. But, if the Jew leaves the house without sealing (at least one seal) the bottle, we are concerned that the goy opened it to drink (we are concerned for both touching and shaking) and is therefore forbidden to drink even if it is expensive wine.
2. Can a Jew pour non-mevushal wine into a glass being held by a goy? There is no prohibition for a goy to drink wine that he touched but the problem is with the bottle. Tosafos quotes 2 opinions whether "nitzok" - the flow of wine, connects the wine in the bottle with the wine in the glass of the goy. We are machmir for any pouring from a small container which is generally not a significant loss. Therefore, if one pours wine into a glass in the hand of a goy, all the wine remaining in the bottle is assur. Even if the goy finished the wine and there is some residue of wine in his glass when the Jew refills his glass, the wine in the bottle will be assur (unless the drops are batul in 60x by the wine remaining in the bottle).
3. If a goy is given a bottle of non-mevushal wine and opens it, is it assur? So long as he hasn't moved the bottle, only touched the outside of the bottle, the wine is still permitted. BUT as soon as he would move the bottle even on the table (machlokes) and certainly if he would lift up the bottle we are concerned that he will shake it for a.z. and it is therefore assur to get any benefit from that wine. If it is an expensive bottle and a significant loss, it can be sold to a goy. It seems to me that it is still forbidden to gift it to a goy unless you will receive benefit in return because the gifting doesn't compensate for loss and would not justify relying on the opinion of Rashi.
4. If one invites a not religious person who would qualify as a "mumar" (such as someone who learned in Yeshiva and publicly violates shabbos), can we give him wine to drink that isn't mevushal? R. Moshe (o.c. 5:37:8) has a teshuva where he permits wine to be given to him even though he will make it assur the moment he touches it. The heter is based on the concept that we have no source in the gemara to consider a mumar like a goy for this purpose and since the prohibition is predicated on the intermarriage concern, it shouldn't really apply to a mumar who is biologically Jewish. Although the custom is to be stringent about this, it wouldn't apply to an uncommon situation such as this.
5. Can one gift a non-mevushal bottle of wine to a co-worker who is a mumar? Rav Moshe's logic would presumably apply to this situation also. Furthermore, since you aren't handing him issur, rather he is deciding to open it at some point later, it doesn't seem to be a torah prohibition of lifnei iver, rather a rabbinic prohibition of helping someone do an aveira which according to the shach in y.d. (based on tosafos in avoda zara 6b) wouldn't apply to a mumar.