And the definitions of element-available() and function-available()
don't say that much, so you can't rely on automatic conversion to
just happen.

Yes, although to be honest I would expect them to happen in this case
(as apparently would implementors of other xslt systems.)

Section 3.2 of XPath 1.0 (function calling) says that the supplied
argument is converted to the required type, which in this case is
string. The second argument of key() is different because the declared
type is "object".

    XSLT uses the expression language defined by XPath [XPath].
    -- http://www.w3.org/TR/xslt#section-Expressions

The very first sentence of section 12 of XSLT 1.0 (Additional Functions)
reads:

    This section describes XSLT-specific additions to the core XPath
    function library. -- http://www.w3.org/TR/xslt#add-func

Can we infer from this that section 3.2 of XPath 1.0, including rules
for type conversion, applies to XSLT 1.0, including XSLT functions?