Looks like what I've been seeing in the stores lately called Opuntia Emerald Wave. Who knows what that means;)?Maybe Opuntia cochenillifera. I've seen that on Opuntias, well I've seen it on some of mine, where the new growth looks dehydrated. Once the pad gets bigger it should plump up.Remember these guys can take the sun. The more sun the merrier for this plant;).

Yup, agreed on the ID. You could probably take cuttings a few nodes long & pot them back into mother plant's pot, to make it look even more full & lush.

"But no one has told me what the difference is between Prunus laurocerasus and Prunus caroliniana. No such clarification was posted. My original question was about identification of a tree. Is there some reason to believe that I have one, rather than the other? " It was properly identified by botanical name by Embothrium. Prior to that, just one of many common names was used for ID however that common name is often applied to both species. Prunus caroliniana is a native species that includes Texas as part of its natural range. It would be very unusual for Prunus laurocerasus to pop up spontaneously in your area but entirely natural for P. caroliniana to do so. It is also more tree-like in habit than Prunus laurocerasus, which is most often grown as a large shrub with no discernible trunk and the leaves present somewhat differently (smooth margins for one and slight serration for the other). As to the negative attributes you list, other than the propensity for P. laurocerasus being invasive in specific areas, they are just not serious enough to worry about.

There are numerous Sempervivums but the webbing points to S arachnoideum.