Measuring area is somewhat easy.
1-The first requirement is knowing what map projection you are using.
I am going to assume you are using an equirectagular map projection. There are no default map projection but this is a very common one to use unless you have something specific in mind.
2- You need to change the projection of an image using a software like G.Projector :
https://www.giss.nasa.gov/tools/gprojector/
G.projector now accepts plenty of different projections as input not just equirectangular.
Pick an equal area map projection such as Mollweide
https://en.wikipedia.org/wiki/Map_projection#Equal-area
It will allow you to measure areas.
ALWAYS KEEP THE ORIGINAL, WORK ON A COPY
3- open the image in photoshop.
On a new layer, paint the area you want to measure in a different colour. Use the pen, not the pencil as you need sharp (pixelated) borders if you want to be accurate.
Using the magic wand and the histogram (windows/histogram) find the total surface (in pixel) of the map (water and earth). Write it down somewhere.
Find the size of the area you are interested in.
Divide the area by the size of the world, this gives you a percentage.
Multiply this percentage by the size of the world in square kilometre, ( or square miles in you live in US or Myanmar) to get the actual size of the area.
For reference, Earth is 510 000 000 km2. So 10% would be 51 000 000 km2.
You don't really need to know the scale of your map, just the % of the world covered and the size of the world.
The find a distance between two points: I think it can be done by using trigonometry where ''a'' is the difference in latitudes, ''b'' is the difference in longitudes and ''c'' is the distance that you are trying to find.
The ''b'' will change depending on the latitude as the distances get shorter the closer you get to the poles.