I get the helicopter after "The chase" mission. I proceed to fly it and land it where other runners do just at the base of the Vercetti Estate Stairs. Then after a few missions such as Supply & Demand... It vanishes?

Why is this? I need that damn heli for COP LAND!

Simply put, don't get into more vehicles than the other runners do and get out/back into your heli after Death Row.

The game only stores your last two vehicles into "Interesting Vehicles" slots (that's how R¤ calls them). Being "interesting" prevents them from being despawned. Getting into new vehicles can replace your heli. Also, there is some bullshit in Death Row. I am too tired to understand all of that, so all I say is that people get out/back in after passing the mission. Details.

Patrick, let me correct you a little bit. Topic you linked is not about why people should reenter heli after Death Row. The reason for it is because game marks vehicle Lance entered as mission vehicle, which removes it from interesting vehicles list, and it is set to despawn after mission ends - unless player enters it after mission again. The topic you linked is about why heli may despawn during mission when player is rescuing Lance. This weird mechanic removes vehicle from said list if any ped wants to pull Tommy out of his car but driver door is blocked.

Anyway, to add to the topic: I understand it this way: helicopter despawns between after "The Chase" and after "Supply and Demand". I see these reasons of despawn:

1. Heli parked too close to Diaz mission marker. Check location in PB runs to compare locations, game may clear some area that removes all vehicles in it and make sure you park it in safe zone.
2. You entered Squalo after "The Fastest Boat". Unlikely but possible - there is no need to do so and it despawns heli.
3. You use more than one car during "The Fastest Boat" to get to boatyard. In runs, only Infernus is used to get there, and even if you destroy it in the process, getting in any other car (not counting boat itself) despawns heli.

As far as I know, if you look down during "Phnom Pehn '86" and see that heli is alright, it should not despawn because of reason 1).

Additional note on similar topic: to keep heli at the mall before Cop Land, you have to use only Infernus from the mall used for teleport until "Bar Brawl" bikers cutscene. If you want to use PCJ from the mission, you have to force game to despawn Infernus is said cutscene. If you leave it too far so that it isn't despawned, or destroy it yourself, getting into any car other than this Infernus will despawn heli. After "Bar Brawl" until detonation cutscene in "Cop Land" you can only use PCJ (or Infernus if it wasn't despawned in "Bar Brawl").