Limit of a sequence (definition)

We don't need that $n_0$ is an integer and it doesn't matter whether we use $\ge$ or $>$ both lead to an equivalent definition.

For the first part of your question, no it does not particularly matter as both statements will hold true for some epsilon greater than 0.

$n_o$ refers to the $n_o$ term of the sequence and thus is an integer. The limit $L$ however can be a real number.