How to prepare gnocchi dough for mixed ingredients?
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.