1. For my case with DS dolls, I remember reading that leaving either the insert or vaginal plug in the cavity prevents the silicone on the top of the opening from developing bends or tears. If the legs are left closed and there is no insert or plug in there, the top of the opening will squish similar to taking a plastic ring and squeezing it until it bends, which might lead to tearing.
I usually also leave the insert in completely cleaned with soap and water and then powdered in case I just happen to want to use my doll for play immediately.
2. This must be different for silicone and for TPE. I have a TPE doll and never leave the insert in. I always insert it before we play and remove it for cleaning after play then put it away.
3. For TPE dolls, definitely remove the insert when she is not being actively used. I had one doll develop a mold infection because I left the insert in her for a week & forgot to set the air conditioner to remove the humidity before I left home. Even if the cavity appears dry, there may be trace amounts of moisture & that is all mold needs to get a foothold.
4. When one of my TPE dolls developed a mold infection in the oral, anal, and pussy cavities, I tried all sorts of different things to try to get rid of the mold. The only thing that worked was to fill the cavities with full-strength bleach and leave it in the cavity for 2 days. This caused some damage to the TPE material; however, the doll was still usable afterward because she used an insert. If the doll had had a built-in pussy, the doll would have been unusable afterward.