Pasta – Why does the pasta I make with the extruder always stick together

You're on the right track! Start adding your flour in the bowl as you did and when you can remove it without it being too tacky then start working it on a cutting board or countertop as you continue adding more flour. Sounds like maybe you pulled it out of the bowl too soon.

As for a specific ratio of flour to potatoes/veg. that's going to depend not only on the items you're using, but also cooking method (for instance I recommend baking potatoes for making gnocchi rather than boiling as they will be drier and need less flour added, thus making for a lighter dumpling).

Once you have enough flour in the mix so that you can knead it and work it, I usually do a "stretch" or "tug" test but holding the dough up in the air and with my hands at each end and then kind of pull and push the dough from the ends (hard to explain in text but imagine the push and pull of two magnets you're trying to push together). If it has a bit of tug and a little elasticity you should be fine. If it doesn't, then either knead it a bit more, adding flour only if it is sticky. If it still doesn't have any tug after kneading more, then you probably don't have enough flour. The gluten in the flour is what's going to help hold it all together so that you don't just end up with disintegrating mashed potato/vegetable blobs in your poaching water. If you're making a big batch it will probably take longer than you expect to get it to the right consistency.

As for types of flour, the more protein a flour contains the more liquid it can absorb. While I have not made gnocchi with bread flour, it would take less bread flour than all-purpose to absorb the moisture in the mix. It will also have a more distinct flavor from the add'l protein. One caveat is that it could end up making them too tough due to the extra gluten.

I have that problem when I am impatient and don't let the pasta dry out long enough before putting it on to boil.

It is important to let the pasta dry even just in loose little heaps once sliced into noodles. It doesn't have to get totally dry just dry enough that you can't squish the noodles back together into dough.

Also you need to cook fresh pasta for a very short amount of time in a LOT of water, about 1.5x as much water as you would use for dry, so that the noodles have less chance to run into each other. In fact I frequently don't boil them at all separately but just toss them in the soup I am serving them with for for a two to three minutes before serving.