I have made sourdough (although starter is sitting unloved in the fridge at the moment). I would not have thought it would work in a bread maker as yeast takes longer to work than commercial yeast and the amount of flour you use while kneading really varies depending on the heat of your kitchen or how "greedy" the dough is being. Geting the recipe right is a bit of a trial and error and depends on the heat and moisture in your kitchen (and the times of day that best suit feeding your bread). I love the bread + reckon making sourdough is worth it if you use at least a small loaf a day AND have 10 mins to spare each morning and evening to feed, knead and bake
I think you have to cook bagel dough for a while in boiling water before baking them. At one time, Jewish bakers were not permitted to use ovens for the time it would take to cook bread, so they hit on the idea of partially cooking the dough in water before baking - or so it said on a TV programme years ago.
I've got a recipe book with a chapter on sourdough breads made with the use of a bread machine, but they are all made in several stages (some over several days) using the machine to make the starter and the dough. The final dough in all the recipes is then taken out of the breadmaker to rise and be shaped and is baked in an ordinary oven.