Do I need to coat beef in flour mixture for slow cooked stew?

Browning your beef with some flour adds depth of flavor. The flour will act as a thickener, and by coating the meat with it you won't have problems with it clumping and getting little flour balls in your stew. However, unless you are browning the meat before adding to the cooker I would recommend you leave it out as uncooked flour might give your end dish a raw flour flavor. You can thicken it up at the end if you like with a cornstarch slurry.

Flouring meat for a stew is a convenient way to thicken the gravy. This tends to work best if you brown the meat with the flour on as it gets the flour properly cooked.

The downside is that it makes it harder to get good caramelisation on the surface of the meat without burning the flour, although for slow cooked stews etc. This is rather subjective and comes down to personal preference.

If you aren't going to brown your meat it may be more convenient just to add a roux (which you can make in bulk and chill or freeze to use as needed). This is better than flouring the meat as the flour in a roux is pre-cooked. You need a fairly high temperature to trigger the chemical changes in the starch which makes it thicken the sauce and slow cookers might not reach that temperature. That would give the dish a raw flour taste and won't work as well as a thickening agent as a roux.

A basic roux is also the base of many sauces and very easy to make.

Some cuts of beef like shin and oxtail produce a perfectly good sauce without flour, especially when slow cooked.

There are also plenty of other thickening agents. I quite like pearl barley in beef stew but peas, lentils and potatoes also work as does tomato paste, but that has a significant impact on flavour (not bad but not necessarily what you want). There are also various flavour-neutral thickeners.

Also, adding a starchy staple near the end of cooking such as rice, pasta, noodles or part-cooked potatoes will thicken the sauce and make a complete one-pot dish.

A thick gravy in stew tends to bring the flavours together well but a thinner broth-like sauce can work as well, especially if you like quite punchy Asian-style flavours.

Flour will help to distribute the seasonings more uniformly over the meat, and they'll stick more easily in the beginning of the cooking process. It will also help thickening the stew later on.

You can probably skip that step, since it's a long cooking time (6 to 10h) and there's no browning in the beginning.

No, you do not need to. As an example, here's a (google translated) traditional recipe on a food site: Matprat (Norwegian site, translated)

It's not properly translated (sos is a local word for sauce (as opposed to saus)), so it's literally called sauce-meat. There are many variations on this recipe, which is not surprising since it's a very simple idea: let meat simmer in sauce until it's delicious.

Variations of this recipe includes browning the meat in flour, simmering with and without vegetables, how thick you want the roux, etc. There's a thousand variations on this simple (and delicious) Sunday dinner dish.

While I have tried both variations the only real difference I have found is that the sauce gets thicker when you make roux AND brown the meat in flour.

I suspect that this may have been part of the reason why people like to flour up the meat before frying. (Part, not all! Other people have made good observations as well!) It gives you a way to get the sauce thicker without adding quite so much butter.

As for the necessity, I would argue that it isn't. If you're going to cook something for a long time, you'll have flavours mixing well regardless, and you have potato and corn flour to thicken the sauce if you feel like it's too thin anyway.