If I have a function f that generates a random number uniformly distributed between 1 and 5 then can I say that g=f*f generates a random number uniformly distributed between 1 and 25?
Fairly obvious not as there some numbers between $1$ and $25$ that you will never get: $7, 11, 13, 14, 17, 18, 19, 21, 22, 23, 24$.
The sum is a bit better behaved, all of $2$ to $10$ will be possible but still not uniform. Consider the similar problem of two dice: $7$ is much more likely than $2$ or $12$.
Also not uniform.
For the sum, the PDF will rise from $0$ at $0$ linearly to a maximum at the mid-point and then fall linearly back to $0$ at $10$.
For the product, also not uniform. I was calculating it but it seems redundant now.