Good answers in terms of meaning. I would add that both навеки and вовеки aren't really used in casual speech.
Навеки is quite formal or bookish and I haven't heard вовеки outside of poems, songs, anthems or religious texts.
Both навсегда and вечно are pretty neutral, both used causally and formally. Their difference is навсегда is more like "from now and forever," as in forever but starting now, or from some point.
Они там останутся навсегда
They will stay there forever (after probably arriving from elsewhere)
Вечно, on the other hand doesn't really imply a beginning, in usage it's more like "always". Something is always like that, unchanged.
Ты вечно молчишь
You're always silent
But if used in the future tense, it could be the same as навсегда
Они будут стоять здесь вечно
They will stand here forever, or they will always stand here (this can mean from now on)