Tasmanian wolf died out around the 1800s.
Q1. Is the above sentence correct? (I'm not sure whether I can use 'around' here.)
Q2. How do I read 'in the 1800s' aloud?
Around works nicely with years and times. But see John's answer to why around the 1800s may not be what you want to use.
To say that a species died out, you can say either
The Tasmanian wolf died out around the 1800s.
Tasmanian wolves died out around the 1800s.
You read or say '1800s' as '18 hundreds' ('eighteen hundreds'). So 'the 1800s' is read as 'the eighteen hundreds'.
When you use the word "around" like this, it means "approximately" or "in the vicinity of" or "in the neighborhood of". When you use it with years, you'd give a rough year as an example:
The problem with "1800s" (pronounced "eighteen hundreds") is that it is a whole century: 1800 to 1899.
As you can see, different people would assume different things when they heard "around" applied to dates, but the 'looser' the date you start with, the 'wider' the "around" suggests.