The other replies have hit the d-limonene based cleaners, so I'll add some and address the acids for the soap scum and lime.
First, use gloves! Anything that will remove the tough deposits is likely to be rough on your hands.
Second, I don't believe kitchen vinegar, which I assume you were using, is a strong enough acid to have a fast effect on the soap scum and lime. (It also won't have a fast effect on the grout, either!) If you want to use a green pure acid product, you probably will have to go to 10% or 20% acetic acid (pickling vinegar and beyond), use the citric acid already mentioned, or go to another organic (that is, carbon containing) acid like glycolic. The thing is, any acid will react some with the alkaline component in most grouts. Because acetic is a weak acid, it still will take longer to loosen the scum and dissolve the lime (and eat the grout) than a strong mineral acid like sulfuric or hydrochloric. Acetic acid is volatile, so it will put off fumes -- try to ventilate the area as best you can. One thing that might help on the doors is to lay them flat, spread whatever acetic acid you're using, and cover the surface with something that will reduce evaporation without being eaten. Plastic comes to mind if you have some used plastic sheeting that you can't recycle, maybe a torn garbage bag or a shower curtain. Once the soap scum loosens, it "should" be removable with a brush and water. As for the grout in the floor tile, you probably can't leave acid on it for a long time (like overnight maybe) without risking some adverse effects on it.
The idea on the "lime," which is mostly calcium carbonate, is to use the acid to pull the lime into solution so you can wash it away. The idea on the soap (most commercial bar soaps really are detergents or mostly detergents, but that's another topic) is similar to "reverse" the solidification (saponification) state and move the components into something of a solution. Acid does both of these; how fast it does it depends on how strong the acid is and how concentrated the acid is. Baking soda and lye work the other way and are of no help to the dissolving action. If you use an acid wash, it would be a good idea to follow it with a baking soda rinse to neutralize residual acid, but a good water rinse probably will work as well. If the idea is to remove the scum by abrasion, baking soda might not be harsh enough -- but it probably won't be too harsh. I've considered using wheat bran or maybe something like rice hulls, but I haven't gotten around to testing them. I'm not too sure about letting a mass of organic matter go down the drain, so clogging is a consideration.
As an aside, I have thought of making a paste of dry citric acid (available in bulk at Whole Foods and probably other places) and water and brushing it on the tiles, but not the grout, and leaving it for awhile. I haven't tried that yet, but it seems like it would work. That might be a bit tedious for the average house cleaner, but it might make a fair lab experiment. As for the commercial soap scum removers, I would not use most of them. I have seen one product that appeared on a glance to be primarily glycolic acid, which is an alpha hydroxy acid that also plays a part in plant and animal metabolism. That might be an acceptable alternative to vinegar/acetic, but I don't remember the ingredient list.
Once one gets into prepared products, it probably takes one mostly back to the citrus-related products. One that I saw that combines citrus oil and organic acids and that sounds pretty good is Heather's Basin, Tub & Tile Cleaner (part of the Jason product family) at:
and apparently in some retail stores.
The ingredient list is: Citric Acid, Glycolic Acid, Salt, Orange Oil, Naturally Derived Biodegradable Surfactants.
I would say that it has become fashionable to add citrus oil/scent to cleaning products, but that sometimes is a cosmetic marketing ploy to mask unfriendly ingredients. I'm not too sure about the Orange Clean Power Paste because I haven't seen the ingredients. The language "orange scented" on the label always makes me suspicious enough to read the ingredient list.