This is true. See, for instance, Theorem 4.3.1 in "Introduction to the Theory of Distributions" by Friedlander and Joshi.

This is true.

By a partition of unity, the proof can be reduced to the case when the test functions have their supports in a unit cube and the result follows from a more or or less straightforward manipultation with the corresponding Fourier series.

See, for instance, Theorem 4.3.1 in "Introduction to the Theory of Distributions" by Friedlander and Joshi (или задачи 423 и 430 в книге Кириллов А.А., Гвишиани А.Д. "Теоремы и задачи функционального анализа").