Use fat-free milk in béchamel

The purpose of using a roux, as opposed to just plain flour, is to improve the dispersal of starch molecules in the sauce. If you just toss a bunch of flour into a simmering sauce - or do the reverse, pour hot liquid onto dry flour - then you'll immediately start getting gross glutinous lumps and will find it nearly impossible to smooth them out.

Starch is not actually soluble in fat; instead the fat coats the starch molecules, making it difficult for them to clump together when incorporated into a sauce. Making a roux out of the flour (and making sure to give the sauce a really good whisking after incorporation) basically helps to ensure an even distribution of starch so that your sauce ends up smooth, not lumpy.

So to answer your question, the minimum amount of butter (or other fat) you need is however much it takes to fully incorporate the flour. If you've still got dry clumps, you haven't used enough fat. More fat is okay, but if you don't use enough then you'll suffer some of the same lumpiness as if you didn't use a roux at all.

I'm not sure if there's a magic number - it's going to depend on the specific flour you're using and the specific fat. Even if you narrow it down to, say, all-purpose flour and butter, it can depend on the brand of flour, the fat content of the butter, and whether or not the butter has been clarified (which, if you have the time and patience, it should be).

The 1:1 ratio (by volume or mass) gives you a pretty decent safety margin; you won't need more than that, but it's also not so much that it'll make your sauce greasy. But you really don't need to measure it out at all; I don't, I just add fat while stirring until the flour is fully incorporated. Sometimes I might be more generous with the butter simply for flavour purposes.

Bottom line, it's really pretty obvious when you haven't got enough fat. If it looks good, it is good, and after a few sessions without the measuring spoons, you should very quickly get a feel for how much fat you really need.

Typically, you go for 1 tbsp flour, 1 tbsp butter for each cup of liquid. You may have to adjust slightly, but that's a good starting point.

After the quantity of roux, the next factor to thickening strength is cooking time. The longer you cook the roux, and consequently, how dark it gets, the less thick the final result will be for a given quantity of roux and liquid.

For a white sauce (i.e. Bechamel and Bechamel-based sauces) you want the roux to stay more or less white. For a Gumbo, you cook it until it's light brown (sometimes called red).