Queries on Rational Numbers

A couple of basic queries on numbers, a part of Math more unfamiliar to me than Geometry. This post is being written with K Sheshadri.

1. Can every rational number h be the hypotenuse of some right triangle of which the base and altitude are also rational? Or are there infinitely many such 'rational right triangles' with h as hypotenuse?

2. a.Given any rational number P, are there infinitely many right triangles having perimeter P and with all 3 sides of rational length?
b. And for any specified rational A are there many right triangles having area A and with all 3 sides rational?
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Query 1 appears to reduce to: "can the square of any rational be expressed as the sum of squares of 2 nonzero rationals?"

And as of now I don't know if this tweak of Fermat's question (last theorem) is done (trivially or otherwise): "No cube of any integer can be written as the sum of cubes of 2 integers. But can the cube of any integer be written as the sum of cubes of 2 (or 3 or k) rationals?"

Updates should follow soon...
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Update (June 12th 2019)

1. Query 1 above has been answered in the affirmative; the same answer leads to an affirmative answer to query 2a (details later).

Another question:
3. Among all general triangles with same rational perimeter and area, are there infinitely many which have all three sides rational as well? (And this same question can be asked with general quadrilaterals instead of general triangles)

Remark on question 3: Consider a thin isosceles triangle with all sides of rational length and area and perimeter also rational (By answer to Query 1 above, such triangles always exist - two rational right triangles team up to form an isosceles triangle). Now, it is not hard to see that this triangle can undergo a continuous deformation that preserves its triangular nature and keeps both area and perimeter constant. Since the change is continuous, all sides change length continuously and so the length of each side passes through a dense set of rational values. However, what is not quite obvious is whether it can be asserted that there are stages in the evolution where all three sides are together rational!

Another way to think of question 3: Easy to see that there are infinitely many triangles all with the same rational perimeter P where are all sides are also rational (one just needs to choose 3 rationals summing to P and also such that two of them sum to more than the third) but among all these, how many triangles also have rational area?

Note added on June 23rd 2019: This post is continued in the next...