Going back to days when starches were first used as routine part of laundering all sorts of things were added to make ironing easier. Butter, lard, candle wax, spermaceti, etc...
Object was to provide some sort of oil or wax substance that would allow iron to glide freely across fabric. This was no small feat back in days when an "iron" was just that, a big heavy piece of cast iron.
If something wasn't put into starch, or even if it was the irons themselves were waxed with some substance for same reasons.
Modern starches for some time now use silicone or some other lubricant that won't stain fabrics, but also will wash out at next laundering.
In theory laundry that is going to be starched by hand or machine should be well rinsed and free of soap or detergent residue. If the starch in question used in bath has any sort of oil or wax yes, it will cut suds to some extent.
All spray starches and sizing also have some sort of lubricant for reasons given above. Again usually silicone and is why directions on container warn about getting the stuff on floors or surfaces. Those oils or waxes will make them slippery.
Main difference between Tide with Downy "Free" and regular Tide "Free and Gentle" is the former contains cellulase enzymes, but not the latter IIRC.
P&G once included cellulase in all Tide detergents (containers had that Cotton registered trademark), but now it seems they pick and choose.
By chewing up bobbles thus making cotton textiles smoother the working theory is laundry will be softer as well. Hence the "Downy" or "Lenor" effect.
There usually is more to it than just the addition of cellulase enzyme. With detergents containing polymers and other substances meant to relax and smooth fibers. Tide Total Care and other products that advertise help protect things from stretching, keep fibers smooth, etc.. all have some version of these systems. Henkel has an entire Perwoll range dedicated to those sort of benefits.