If you leave it alone it will start to climb that pillar. It is a creeper which becomes a self adhesive vine when it finds a vertical surface. The leaves are normal sized. It's classed as an invasive species in some states, I believe.

Something like Euphorbia hypercifolia 'Inneuphe'. In the US it is sold under the name 'Diamond Frost'

there is a houseplant forum .. for further learning ... if it had been taken outdoors in my MI ... 3 weeks ago ... it would have a lot more problems than yours .. cold damage ... also.. its stressed from the change of circumstances.. e.g. going from the growers greenhouse ... to the seller.. and to your house ... so a little leaf loss is not that big a deal ... also .. in fall.. many plants lose their older leaves ... due to the changing season ... anyway ... its stressed.. and not looking all that bad ... just water it properly .. and it should put out some new leaves ... and being stressed.. does not mean its hungry .. fert is not a response at this time ... if you just got it from the store.. i am sure there is enough fert in that pot for months ... in spring .. when the growing season starts.. then some fert might be helpful ... ken ps: its not suffering.. its just reacting to change .... you are suffering.. try not to love it to death ...

Yes, usually rot. You can just feel it and the color is very much telling too...but if unsure, take a branch and start cutting thinner slices from the bottom to see what the flesh looks like. If very soft, you may not be able to get clean cut. Use a blade, scissors or pruning shears will just squish it. If you find firm flesh, there is a chance that it could be rooted - few of the stems may have some viable parts. The longer, thicker stem on left seems to have yellow tip too -BUT very light color could be caused by plant being shaded too much for a while. I am assuming rot since the main stem was rotted, and being able to see only photos.