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1. As an Evolutionist, is it your belief that evolution is a continuous process? That is, is it a process that has continued at roughly the same rate since the start of the Pliocene epoch (the last 13M years)?
...a. If so, what species is now most closely "related" to man?
...b. If not, then when was the last period of "punctuated" change?
......i. What natural influences cause periods of "punctuated" change?
......ii. What now extinct species is most closely related to man?
......iii. In interim periods, is the evolutionary process halted, or simply slowed?
..........1. If slowed, would you ascribe race as the evidence of gradual evolution?

2. As an Evolutionist, how would you characterize the next species that will evolve from man? (either gradually or after a period of "punctuation.")
...a. Do you expect that more than one species will evolve from man?

3. As an Evolutionist, what determines that an organism's species has changed?

You assume that the periods of punctuated equilibrium are always global catastrophes. No. One species can be in trouble while another is doing just fine. The one in trouble may have broken up into small sub-populations, some of which may fairly rapidly evolve a new adaptation, some of which probably go extinct. The various species that are doing fine will probably stay about the same, since the pressures on them are to basically not break what's working.

The now-extinct species most closely related to man might be archaic Homo sapiens, assuming it would even be a different species. I'll avoid Homo sapiens neandertalensis since it's generally not classified as a distinct species, although that may change. Anyway, Homo erectus is the ancestor of both of the preceding and is generally considered a separate species.

Some species don't show any morphological change in practically forever. Cyanobacteria (blue-green algae) are a case in point. They presumably have mutations, but anything that takes them far from their adaptation dies out. They're at a local fitness maximum.

2. As an Evolutionist, how would you characterize the next species that will evolve from man? (either gradually or after a period of "punctuation.") ...a. Do you expect that more than one species will evolve from man?

Evolution isn't teleological. All one can really say is that the next species evolved from homo sapiens will be related. As to whether more than one species will evolve depends mostly on whether there is some form of spatial separation between breeding groups.

3. As an Evolutionist, what determines that an organism's species has changed?

"Species" is not a property of an organism. It is a (rather complicated) relationship among organisms. To some extent, the taxonomical terms (species, genera, family, etc.) are arbitrary. One can draw a tree of relationships and make some judgments about nearness. The entire field of cladistics (several journals) is devoted to these questions.

Don't expect any answers to your well thought out questions by the evolutionists, they don't answer questions. You see, evolutionism is some kind of Eastern religion which sutstains itself by chanting mantras such as "God did not do it", "Evolution is true and anyone who denies it is an idiot", and "Darwin was a great scientist". They also have an idol they pray to, it is a bent over monkey with a beard.