There are assortedmany types of people in the world.

Assorted tends to mean a variety collected in one specific place. ("Assorted chocolates" in a box of many chocolates (Truffles, caramels, nuts). "Assorted tools" in a specific tool box (hammers, screwdrivers, etc.). "Assorted people" for a job that requires many skill types (carpenters, painters, plumbers). )

She is a perceptive girl since she can easily feelunderstand other people.

Your use of 'perceptive' is correct.

I huffed and puffed because I wanted to catch up to my bus.

Biologists are those who know a lot in theiry is a biologist's province.

Most people think of "province" as a physical territory. This word is less known in English, but you are correct.
"Wine is a sommelier's province."

We need to map out a plan for bolstering our economy.

How can they can escape unscathed from this accident?

There are assorted people in the world.

We don't use "assorted" with people. Usually we use it for things such as sweets/candy, colours, etc.

She is a perceptive girl since she can easily feelread other people.

"feel" here seems weird

I huffed and puffed because I wanted to catch up to/with my bus.

Noxious fumes damage our bodyies.

Biologists are those who know a lot in their provincefield.

Province is a political term used for denoting a large area.

We need to map out a plan for bolstering our economy.

How could they can escape unscathed from this accident?

with question words the verb comes before the subject. This sentence sounds like its in past tense