Do these apply today?
The simple reading of the Gemara in Bava Kama 38a seems to say no. The Gemara says:
מה ראה ראה שבע מצות שקיבלו עליהם בני נח ולא קיימום עמד והיתירן
The Gemara then asks how could that be, the sinner is profiting? The Gemara answers that even if they are מקיים them they only get שכר כאינו מצווה ועושה.
Rashi there seems to take the Gemara literally that they are no longer חייבים. Others understand the Gemara that they still are חייבים however, even so, they don't get שכר like a מצווה ועושה.
According to Rashi, if they are not חייבים how can they be punished for not keeping them? The Netziv answers that the ז' מצות בני נח are really mitzvos for the good of society. Without these miztvos society cannot survive. Therefore, if people violate these mitzvos they are not being punished for violating them, rather society will fall apart and they will suffer because of their actions. The Meshech Chochma (beginning of וזאת הברכה) says a similar idea that the ז' מצות בני נח are for societal order. Since Hashem built the world based on the Torah, whenever the ז' מצות בני נח are not observed things don't work well.