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If the country is not increasing in population growth rate, what are some socio/economic problems that might occur?

In such a situation, the average age of the population increases and, unless there is an actuarily sound retirement funding program (instead of "pay as you go", like our Social Security) fewer workers must pay for the retirement benefits of an increasing fraction of the population. This increases their tax burden and limits economic growth. There will also be a problem with a labor shortage in low-skilled service sector, making it necessary to use foreign workers or illegal aliens in low-paying jobs. Otherwise, the standard of living will fall.

oh! I see.

what exactly does increasing/decreasing population growth say about the country?

also, if birth and death rates remain the same, what is the population pyramid going to look like in a couple of decades?

This is not the kind of "science" question I expected. My area of expertise is physical sciences. Perhaps someone else can help you if you repost the question with the subject as "economics" or "social studies" or "demographics"

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