Which language class are today's modern programming languages like Java, JavaScript, and Python in?
It appears (?) they are not context-free and not regular languages.
Are these programming languages context-sensitive or decidable languages? I am very confused!
I know that context-free is more powerful than regular languages and that context-sensitive is more powerful than context-free.
Are modern programming languages both context-free and context-sensitive?

Practically no programming language, modern or ancient, is truly context-free, regardless of what people will tell you. But it hardly matters. Every programming language can be parsed; otherwise, it wouldn't be very useful. So all the deviations from context freeness have been dealt with.
What people usually mean when they tell you that programming languages are context-free because somewhere in the documentation there's a context-free grammar, is that the set of well-formed programs (that is, the "language" in the sense of formal language theory) is a subset of a context-free grammar, conditioned by a set of constraints written in the rest of the language documentation. That's mostly how programs are parsed: a context-free grammar is used, which recognises all valid and some invalid programs, and then the resulting parse tree is traversed to apply the constraints.
To justify describing the language as "context-free", there's a tendency to say that these constraints are "semantic" (and therefore not part of the language syntax). [Note 1] But that's not a very meaningful use of the word "semantic", since rules like "every variable must be declared" (which is common, if by no means universal) is certainly syntactic in the sense that you can easily apply it without knowing anything about the meaning of the various language constructs. All it requires is verifying that a symbol used in some scope also appears in a declaration in an enclosing scope. However, the "also appears" part makes this rule context-sensitive.
That rule is somewhat similar to the constraints mentioned in this post about Javascript (linked to from one of your comments to your question): that neither a Javascript object definition nor a function parameter list can define the same identifier twice, another rule which is both clearly context-sensitive and clearly syntactic.
In addition, many languages require non-context-free transformations prior to the parse; these transformations are as much part of the grammar of the language as anything else. For example:

Layout sensitive block syntax, as in Python, Haskell and many data description languages. (Context-sensitive because parsing requires that all whitespace prefixes in a block be the same length.)

Macros, as in Rust, C-family languages, Scheme and Lisp, and a vast number of others. Also, template expansion, at least in the way that it is done in C++.

User-definable operators with user-definable precedences, as in Haskell, Swift and Scala. (Scala doesn't really have user-definable precedence, but I think it is still context-sensitive. I might be wrong, though.)


None of this in any way diminishes the value of context-free parsing, neither in practical nor theoretical terms. Most parsers are and will continue to be fundamentally based on some context-free algorithm. Despite a lot of trying, no-one yet has come up with a grammar formalism which is both more powerful than context-free grammars and associated with an algorithm for transforming a grammar into a parser without adding hand-written code. (To be clear: the goal I refer to is a formalism which is more powerful than context-free grammars, so that it can handle constraints like "variables must be declared before they are used" and the other features mentioned above, but without being so powerful that it is Turing complete and therefore undecidable.)
Notes

Excluding rules which cannot be implemented in a context-free grammar in order to say that the language is context-free strikes me as a most peculiar way to define context-freeness. Of course, if you remove all context-sensitive aspects of a language, you end up with a context-free superset, but it's no longer the same language.

