No you don't have to do the 15 in time to get the weekly chest. Just participate in doing a M+ and you will get rewarded with one :)
I'm part of a guild that raids on the heroic level, but I am just a spectator. Not part of the raid team. I don't raid at all. I do, however, do my +10 or +15 mythic dungeon every week for the chest reward. I have higher ilevel gear than almost everyone in my guild who are doing heroic raiding. The chest reward outclasses people who only do heroic raiding by quite a lot, especially as the weeks go by and I can catch up to their initial week 1 burst of new loot.
you don't have tiers in M+...It;s not everything in ilvl but the quality of the items...