TRENDS — There are lots of ways to get evolutionary trends. Standard organism-level processes are one. Species selection is another. I discuss this in this paper (free access).

SPECIES SELECTION — No, this does not depend on punk eek. Gould talked about the possibility under more gradual speciation in The Structure of Evolutionary Theory as well.

SEX — The important thing about major features is that there are four questions, which may be answered differently: 1) Why did it arise (e.g., which mutations)?, 2) Why did it reach a high relative frequency initially (microevolution)?, 3) Why is it maintained in certain lineages?, and 4) Why is it distributed as it is among lineages?
Species selection would deal with question 4.

“Wright’s rule” is the speculation that “speciation might be truly stochastic with respect to evolutionary trends”.

Does stochasticity in speciation with respect to evolutionary trend depend upon whether a lineage shows a pattern of punctuated equilibrium?

If it doesn’t then Wright’s rule may establish conditions for species selection regardless of whether the lineage shows gradual evolution or a pattern of PE.

So Wright’s rule + the trait undergoing the trend affects rates of speciation + natural selection on individuals not opposing the trend should be sufficient conditions for species selection to form trends. (if selection within populations for example favors small bodied individuals then a macroevolutionary trend of increasing body size in species won’t form). None of these depend upon increased rates of evolution during speciation.

Is there need to invoke PE as a necessary condition?

That's because punctuated equilibria by itself cannot explain species selection. In a nutshell, the argument for species selection from punctuated equilibria is summed up in the 1977 Gould & Eldredge paper:

punctuated equilibria + Wright’s rule = species selection

"Wright's rule" is the speculation that "speciation might be truly stochastic with respect to evolutionary trends". This means that microevolution and macroevolution are "decoupled", and a species is a good "Darwinian individual". Essentially, speciation is random (and analogous to mutation) and species are analogous to alleles. Competition between species leads to species selection and this can explain evolutionary trends.

Take a look at Figure 5-10 in the original Eldredge & Gould paper from 1972 (also reprinted in Gould & Eldredge 1993) and you'll see exactly what they're talking about.

Traditional evolutionary theory and punctuated equilibria can both explain trends, but they differ in HOW they explain those trends. Gould liked species selection as a means of explaining evolutionary trends without any reference to orthogenesis or evolutionary 'progress'.

With reference to sex, the problem is that sex can't easily get off the ground because there are short-term disadvantages, mainly:
1) sexual recombination breaks up favourable genotypes
2) a sexual female will have only half as many grandchildren as an asexual female (called "the two-fold cost of sex"), and so genes for sexual reproduction should be eliminated by natural selection

Species selection is one way around the problem. i.e., Sexuality evolves rapidly in a small peripatric population, and this species then goes to outcompete others, resulting in the takeover of sexuality. (Incidentally, this answer to the problem is almost certainly incorrect.)

Finally, he suggests that sometimes a feature can be subject to selection at a higher level that is not impeded at the lower level, thereby making it widespread among extant lineages (such as sexual reproduction, which delays extinction and accelerates speciation).

But Sexual species will proliferate and become more common than asexual species regardless of whether they originate gradually or rapidly. And the same reasoning can applied to other macroevolutionary trends, example a trend towards larger body size. Some population level trait makes one type of species more numerous than some other type over time.

does that depend on whether the species shows gradual change within its lifetime or whether change is concentrated in rapid bursts?