need help with ninja

Just hit 46 and got my full arsenal of abilities (except kasatsu).

First problem:
rotation. I cant seem to maintain all debuffs and dots continuously without having downtime or overlaps. This is especially so after applying ninjutsu, which cause different delay to the timings and the movement needed for sneak attacks. I guess my question is more like, is it possible and are we expected to ensure all debuffs and dots are applied at all time?

Second problem:ninjutsu lags (or more like bugs?). My ninjutsu animations go off without the skill actually going off...

Example: if I try to use futon and click through ten>chi>jin rapidly and click the ninjutsu button, very often the RAITON animation would go off (as if jin was not applied). Naturally, I would continue with attacks but once I use a skill, the rabbit medium icon appears at the ninjutsu button. Basically, the skill didn't actually go off. In the same scenario, if I keep mashing the ninjutsu button, futon animation would go off after ration animation, only that this time futon was actually used. This happens when I try to use Raiton too (then fuma shuriken animation appears when the skill isn't actually used).

I reckon it is latency but just want to confirm with guys.

This is causing me to mess up rotations even more because I now have to wait and observe (which takes a while) to ensure that the correct ninjutsu is actually used, and making chaining dots and debuffs seamlessly almost impossible.

Same happens for me (Im in japan so I think its lag on my part) but I find the best rotation after your opener should be dancing edge (10% slash resist down) then shadow fang then mutilate. Always prioritize huton and when you get it Trick Attack. Other than that keep the dots refreshed as best you can. I have times where 2 dots have the same timer so refresh the one that takes longer to get to Ie. Dancing>Shadow>Mutilate. There are plenty of guides around if you need more help.

For your first problem, prioritize keeping Huton up but don't waste a Ninjutsu refreshing it if you have plentry of time left on your buff. Use Trick Attack whenever it is off cooldown. Keep Dancing Edge up constantly unless you have a WAR tank providing the slash debuff consistently for you so you can focus on Aeolian Edge instead. Lastly, keep up Shadow Edge and Mutilate when you are able. With pracitce, you should be able to keep everything active nearly 100% of the time.

While Huton is active, use Raiton if you have good latency and especially if Foe Requiem is being used. If latency is causing issues, use Fuma Shuriken as it will be buffed by Dancing Edge and only needs a single mudra.

For your second problem, it is a known issue. It isn't a bug, it's just lag which is really pesky for some and not so bad for others. For me, I don't have issues most of the time and can fire off mudra no problem. For others Ninja is relatively unplayable because mudra are unreliable with the lag. Hopefully it will be addressed sometime soon-ish.

Well, I have some lag too but I don't think you need to wait for the ninjutsu icon which is not always reliable.
Sometimes the Ninjutsu action button doesn't work if I press it once, like in the OP.

I just do my mudra in order and press the ninjutsu button a few times. Then I queue up the next action.
Ninjutsu still goes off and the next attack goes off soon after while both the ninjutsu icon and character animation catches up.

It still eats into the GCD, but waiting will take even more time.
You can press the ninjutsu action a few times with no consequence. It's not wasted time either since a few presses is faster than finishing the ninjutsu, so my next action is already queued before it's all done.

For keeping everything up, it's not so bad even with lag.
The lag is just annoying because it feels slower than it should.
I refresh Fuuton if my ninjutsu is up and the buff has <20s.

The rest is as Pun and Dark has said.
I'm fairly new too but I've already gotten a feel for when my dots/debuffs need refreshing without really monitoring them actively.
I used to main a Monk whose timers are mostly built into the combo.
NIN is asymmetrical but still has a rhythm (like DRG, I guess).

Latency shouldn't cause downtime unless you mess up a Fuuton cast because of it.
Just do the combo in order and smash the ninjutsu button a little and it shouldn't happen though.