Should non-income-producing activities be considered productive (in terms of virtue)?

The question states that "only activities that produce income are considered expressions of the virtue of productivity." The question attributes this view to OPAR, but does not cite any specific references.

I checked OPAR and found productiveness described in detail in Chapter 8. The section on productiveness begins:

"Productiveness" is the process of creating material values, whether goods or services. Such creation is a necessity of human survival in any age, whether the values take the form of bearskins, clubs, a pot full of meat, and paintings on the walls of caves; or of skyscrapers, ballet, brain surgery, and a gourmet meal aboard a computerized spaceship; or of the unimaginable luxuries and splendors yet to come.

The key is creation of material values. This is "sort of" like "activities that produce income," but not exactly, not so narrowly. A short while later, that section of OPAR eases into a discussion of "productive work," which is treated as essentialy synonymous with productiveness. OPAR closely follows Ayn Rand's TOE in this discussion, which likewise treats productiveness and productive work as basically just different emphases on the same activity. For the essential excerpts from TOE and also Galt's Speech (GS), refer to the topic of "Productiveness" in The Ayn Rand Lexicon.

The question also states: "since all the cardinal virtues are conceptual subdivisions of the virtue of rationality, each of them contains all of the others." The first part of this formulation matches the Objectivist view as stated in OPAR, TOE, and GS. The second part, however, does not. The mere fact that two (or more) subsets come from the same larger set does not mean that the two (or more) subsets overlap in whole, in part, or to any degree at all. The beginning of Chapter 8 in OPAR makes the relationship clear:

[Since] the derivative virtues (and values) recognized by the Objectivist ethics ... are expressions of rationality, they are logically interconnected, both in theory and in practice. None can be validated in isolation, apart from the others; nor can a man practice any one of them consistently while defaulting on the others.

This does not say that each of the virtues other than rationality "contains" all the others, nor that "every rational act is at once an expression of integrity, honesty, justice, etc." The other major virtues are certainly aspects of rationality, but it is an unwarranted leap to treat them as aspects of each other. One should not interpret OPAR (or TOE or GS) as claiming that "productiveness" has to be found in "every rational act." It can be entirely rational just to relax from time to time, or enjoy a work of art or a fine meal or the companionship of a romantic partner, and so on, without implying that one is (or must be) engaging in productiveness or productive work in so doing. Enjoyment of life is certainly made possible by practicing productiveness and all the other virtues, but this must not be taken to mean that one must engage in productive work during every rational moment of one's life.

A follow-up comment by the questioner mentions "Branden," evidently referring to Nathaniel. If the question's misunderstandings of what productiveness is and how it relates to the other virtues are influenced in any way by anything written by Branden, I would advise limiting any study of those works to articles published originally in The Objectivist Newsletter or The Objectivist, at least until one has completed a thorough study and understanding of GS, TOE and OPAR. I would also rank Tara Smith's books on ethics far ahead of Branden's later works (post-split) as elucidations of ethical concepts and principles.

I wrote up my thoughts on parenting as a productive endeavor in a blog post today.

In short, I think parenting is productive because the activities involved are actions taken to sustain a material value in the world (the child). Children need adult guidance and support (physically, cognitively, psychologically) and a parent who provides for those needs is working productively--and selfishly.

I can't recall anywhere where Rand or Peikoff states that productiveness must result in an income or potential income.