So I sided with Saadia and killed the Alik'r in "In My Time of Need." Now two Alik'r warriors have parked themselves just inside the Whiterun gate. Apparently, waiting for Saadia to try to leave the city. They're not hostile to me, nor are there any dialog options. How long do they stay there?

Unfortunately they are not removed, unless you use the console on PC, or until the attack/defence of Whiterun in the civil war questline. At least that's as far as I know. You could always try to kill them though.

They're there forever. You'll also find random encounters along the road, where couple of Alik'r warriors will be pestering random redguard woman.

They always wish me a good morning, nice day, and so on. I don't mind them to be honest.

If you kill them (or if they fall victim to a random Werewolf attack) they seem to stay gone. They didn't return to my one game after ingame months, but I've since completed the Civil War so IDK if they would've returned given enough time.

Long after I completed the quest, I was randomly passing through Whiterun gates, ran into one of the Alik'r... "Here, friend" he gave me some potion as a gift.

Killing them doesn't turn the Whiterun Guards against you or incur a bounty?

thar there FOREVER! Nwaahaahaa! Nah, i dunno, but they seem to be, they don't leave, sleep, nothing...

I took that route of saving saadia becos i herd she opens up a shop in which she surely does but guess what she's selling? and how much money she has??
GAH!

I had a slightly similar problem on one of my old characters, when saadia took me upstairs and threatened me with a dagger, i quit the chat and decapitated her, then 2-3 alikr warriors just stayed at the whiterun gate (until i killed them...)

They are hanging around for me on three characters, though to be honest, I can't recall doing the quest on my latest guy. I did, however, clear out that hideout a day or two ago. According to the wiki, it's another wholly ambiguous quest, where you can't be sure who's right or wrong. There is a way of getting rid of both parties, but I find it hard to side with the warriors. Their constant harassment of every redguard gal in Skyrim is almost as bad as the Thalmor's terrorism. Besides, Saadia provides prompt service at the Bannered Mare, lol.

Nope. Saadia is lying to you. Hammerfell's been at peace for 20 years after the redguards kicked the thalmor out.(Source: http://www.imperial-library.info/content/great-war) Saadia says the thalmor are still fighting the redguards(Definitely wrong). The Alik'r say Saadia betrayed them during the war.(Possibly wrong) The Alik'r are an actual military force of the Redguards.