Yes, hextech after rageblade is probably the best build path right now, however if the enemy team has a fed AP, you usually want a Hexdrinker before going for gunblade.
You can definitely do that, especially when you think that no fight will happen in the first few minutes ( farm champs). However, you'll not be able to gank/counter gank top or mid safely, should a fight happens there early since you are low on health by the time you reached top side
Special thank in "What to do during each phase of the game" chapter.

Death's Dance is not very good right now, however, if the item is buffed in the next patch, i'll definitely give it a try. I also agree that not having lifesteal/spellvamp is a problem.
Updated my the warmorg buid, with special thank to you in item explanation chapter.

I agree on rushing rageblade before hextech because of the damage increase with his passive, but hextech is great after rageblade imo. Btw when I play Jax, I can get Stalker's blade on my first back and not have to make any more trips back for it.

Gunblade is a nice item, but it doesn't provide a massive power spike like rageblade would. Gunblade is strong for its lifesteal passive while doesn't provide big dmg spike. I also don't rush trinity, just a sheen, it's for some extra burst to make gank and clear easier while getting the gold for rageblade. However, with recent testing, it seems that the sheen rush is sub-optimal.
Nashor is actually a really good item, now that i think of it, i'll include it in my guide, with your name under it , of course.

So. Don't rush Hextech?. I mean, I always rush the Gunblade (by buying as first item the Cutlass) but now I'm a little bit confused.
And I buy the Nashor's Tooth too instead the Trinity... can you explain me why buy Trinity first and why you don't buy the Cutlass as first item? :P