Does a series have to start from 1 to be able to apply convergence tests? In the ratio test, for example, we know that the test applies for a sum from n=1 to n=infinity.
Can I directly apply this test to sums from n=x to n=infinity (for example, a sum from n=0 -> infinity)?

You can always write your sum as a sum starting with the index $1$ by performing an index shift:
$$
\sum_{k=k_0}^\infty a_k = \sum_{k=1}^\infty a_{k_0+k-1}.
$$
