What it checks
The mechanical rules first: names must be short and restricted to letters, digits, underscores and hyphens, and schemas must be objects with properties. Breaking either is an API rejection.
Then the ones that do not error but do cost you: a missing description leaves the model with no basis for calling the tool, a very short one leaves it guessing, and a 400-character one is billed on every single request forever.
The serialised size
Reported explicitly because it is the number that matters and the number nobody sees. Tool definitions are fixed overhead: paid on every request, on every iteration of every agent loop, whether or not any tool is used.
If the figure surprises you, the usual causes are prose descriptions of parameters the schema already declares, and deeply nested optional objects.