Refined Goldbach conjectures for selected moduli

Let $p$ and $q$ be primes. The selected-moduli refined Goldbach conjecture. The following assertions hold:

1. Every even $n\equiv0\pmod3$ except $n=6$ has a representation with $p\equiv-q\equiv1\pmod3$.
2. Every even $n\equiv0\pmod5$ has a representation with $p\equiv-q\equiv2\pmod5$; apart from $n=10,20$, it also has one with $p\equiv-q\equiv1\pmod5$.
3. Every even $n\equiv0\pmod7$ has a representation with $p\equiv-q\equiv3\pmod7$.
4. Every even $n\equiv0\pmod{11}$ has a representation with $p\equiv-q\equiv3\pmod{11}$.
5. Every $n\equiv0\pmod8$ has a representation with $p\equiv-q\equiv3\pmod8$.
6. Every $n\equiv0\pmod{16}$ has a representation with $p\equiv-q\equiv3\pmod{16}$.
7. For any fixed integer $a$ coprime to $60$ and not congruent to $\pm1$ or $\pm11\pmod{60}$, every $n\equiv0\pmod{60}$ has a representation with $p\equiv-q\equiv a\pmod{60}$.

These are explicit Goldbach-type conjectures based on computations; only the first item is described as a direct consequence of binary Goldbach. The remaining assertions are supported by numerical evidence but are unproved.