Can new brain cell development slow the progression of Alzheimer's?

It is not clear that rigorous mental work, of the types you described, actually leads to the development of new brain cells. However, it is well established that new connections are formed between existing brain cells when new information is learned. Additionally, old connections get strengthened when used to recall previously stored information. Given that more connections are better than fewer connections, and strong connections are better than weak ones, activities of both learning and recall are likely to benefit brain health.

In terms of the relative benefits of rigorous brain activity compared to the detriments of damage caused by Alzheimer's disease, it is too complex and too individual to say for certain how much one can slow down the disease. Some people, with certain genetic profiles and overall good health might be very effective at slowing the disease with diet, exercise, drugs, and mental stimulation while others might not fare as well.

One form of mental activity that is sometimes overlooked is mere socializing and forming new friendships. Meeting new people, sizing them up, and performing mental accounting of their character traits is something we all do quite naturally and requires robust mental activity.

For sure, there is no harm in keeping your brain active and engaged as a means of warding off mental decline due to Alzheimer's disease.