The rate of people living in poverty in Europe was fluctuatinged between the years 1993 and 2005.

If a specific duration is mentioned like you've done, past indefinite tense is all that is required to indicate that the action happened over that time period.

Until the year 2015 when the chart ends, we can observe a downward trend as the number of poor people has been dwindling.

This simply clarifies that the downward trend ends in 2015 because the chart ends rather than the trend ends because of some social/economic or other reason.

As for the number of the poor people in the Middle East and South Africa it flatlined from 1999 untill 2013 after when it started toich it escalated and reached about 6 million poor people in 2015.

A slightly more aggressive correction could be "As for poor people in the Middle East and South Africa, the number flatlined from 1999 until 2013 after which it escalated and reached about 6 million poor people in 2015."

“After which” means the same thing as "and then," but it sounds formal/technical sounding than "and then."

The global number of PC and smartphone users has mounted to 1700 million people in 2014.

Global population has been steadily escalating sincetarting from 500 BC and untill 1500, after which it started to soaring and equalled 4000 m billion people in 1979.

since->starting from, this is mostly stylistic correction. "starting from" is much more commonly used.
4000 million -> 4 billion, this may be a difference in long/short scales between English and your native language see:https://en.wikipedia.org/wiki/Long_and_short_scales

The upward trend has been continuing and it is forecasted that the number of people inhabiting our planet will surpass 8500 m.5 billion people in 2025.

same as above