How to use vanilla beans? I used a vanilla bean in a water-based beverage and did not get any taste.

Further research showed that vanilla is apparently oil soluble, not water soluble.

I know that it is a common practice to make vanilla tinctures using alcohol, but if it is fat soluble, why not use oil? The problem with alcohol is that it gives bad taste to many things.

I see in vanilla ice cream there are the little black seeds. Do they just mix the seeds in the milk, and the milk fat absorbs the taste? How long does this take? If I want to add vanilla to a fat-based food, how long do I need to let the seeds sit in it for the flavor to spread throughout?
Vanilla extract is essentially vanilla infused into alcohol. If you don't have any problem with storebought vanilla extract, I doubt you'd have much problem with homemade vanilla infused into alcohol (i.e. homemade vanilla extract) either. While people do sometimes describe real vanilla extract as "boozy", it's not so much about the alcohol as the vanilla flavor. Some do still like it a bit less, though.

Infusing into oil will certainly work, but it does introduce botulisum concerns if you want to store it for very long. So you're much more likely to see it infused directly into a high fat food like ice cream.

For ice cream, you scrape the contents of the bean into the warm custard mixture (milk/cream/egg), and let it steep for 20-30 minutes. (You can also put the whole bean in and fish it out later.) The heat and the fat content both help extract the flavor.