Let's ignore 'triangles' with sides of length zero. I assume that you mean triangles with sides of whole-number length; otherwise, the possibilities are infinite.
The sides have to add to five. There are only two ways of doing this:
1 + 1 + 3
1 + 2 + 2
Both of these possibilities satisfy the other constraint mentioned in the question.
It is an isosceles triangle with 2 equal sides.
Do exist.
There is one equilateral triangle with 3 equal sides of 7in
A triangle with two equal sides is an isosceles triangle and you can have as many as you like.
Sausage roll
There is only one triangle. Or two if you count its mirror image as a different triangle.
How many triangles exist with the given side lengths 3in, 4in, 2in
Yes, they do exist. And the question is ... ?
3
Infinitely many. Every triangle can tessellate and each will result in a different tessellation. Since there are infinitely many possible triangles, there are infinitely many tessellations.
More than one unique triangle exist
There is only one.