TypeScript does allow the with statement, as not doing so would violate its being a superset of JavaScript. However, the compiler will emit an error (though still compiles and outputs code), because there is no way for anything to know anything about what is being referenced from a with statement except at runtime, which completely defeats the purpose of static typing, so you shouldn’t be using it in a TypeScript program. There’s an additional discussion on the TypeScript forum about this.
With regards to other strict mode things not allowed in TypeScript, since TypeScript tries to follow the ES6 specification where it can for its new features, and the ES6 specification (§10.2.1) says “Module code is always strict code.” and “A ClassDeclaration or a ClassExpression is always strict code.”, you will probably find that nearly all your TypeScript code will need to conform to strict mode at some point in the future, if not today.