What people said above. Also to answer in order :

1) Does savage have "seasons"?
Kinda. On patch X.X, the tier drops.
on X.X1 or X.X1 (one or two weeks later) some adjustements are made to jobs or boss if a real imbalance exists.
on X.X5 some other adjustments for minor imbalance between jobs based on their study of clear data
on X.X8 the limit is lifted
on X.X+1, the echo is added

1a) Why isn't there a phased "stat boost" for the community via echo for handling savage after the first six to eight weeks?
Because it would then be a race for midcore players who want to tackle it (and show they do) without echo. Also by progessively gearing in the tier you already boost a bit, in case you can't manage the checks in crafted gear (the difference between a carfted pentameld set and a non BiS but max ilvl set is a 10% boost in itself)

2) Is unlocking savage linked to it becoming old content, or is unlocking it a catchup mechanic for people who are behind or just do not want to have to deal with clearing the first three fights?
It is both, also you'll notice their cycle is purposely designed to NOT make any content irrelevant from patch day. If they ease Savage, they would kill the need for the next Ex primal and the alliance raid, depopulating those contents from even day 1. So they carefully, progessively add new ways and so put a limit on the existing way. Have you noticed how you can currently upgrade armor and accessories but not the weapon with alliance/hunts, and only will able to do so in 6.38 ? Because if they didn't Rubicante Ex or the relic would have had less appeal to gear secondary jobs.

It's a whole lot of small stuff like that but when you pay attention you see how they do care about content "relevance" and how it affects nerfs/limitations on accessible improvements.

3) Why is it setup such that after someone clears savage, they still have to complete the entire sequence in order?
To populate PF on lower stages even if you only need last stage