To suggest an idea on what you asked:
- Make a vehicle path for the plane, pretty sure there is a ugx tutorial on it
- set variables on the level for the next plane round
- use a projectile impact check on the plane when flying

#include common_scripts\utility;
#include maps\_utility;
#include maps\_zombiemode_utility;

plane_round(planeDamage){
while(1){
	if (level.round_number >= 3){
 
	//level.round_number >= 15 && level.round_number % 2 && planeDamage < 3
 
		iPrintLnBold( "Round "+level.round_number+". This is a plane round" );
 
		plane_pass_by();
		planeDamage = plane_hit_detect(planeDamage);
	}
}

plane_pass_by(){

	plane = getEnt("plane","targetname");
    plane_path = GetVehicleNode("plane_path_start","targetname");
    plane AttachPath( plane_path );
    plane thread maps\_vehicle::vehicle_paths(plane_path);
    maps\_vehicle::vehicle_init(plane);
 
	plane StartPath( plane_path );
 
	iPrintLnBold( "Plane flying over the map..." );
 
}

plane_hit_detect(planeDamage){

	plane = getEnt("plane","targetname");
 
	/*while(1){
		plane waittill( "damage", amount, attacker, direction_vec, point, "MOD_PROJECTILE" ); 
		iPrintLnBold( "Plane has been damaged" );
	}	*/

	if (planeDamage > 3){
	plane_down();
	return 4;
	}
	else{
	return planeDamage+1;
	}
}

plane_down(){
 
	iPrintLnBold( "Pap falls from the sky!" );
}

plane_pass_by(){

I always call my stuff in mapname.gsc with the init() function for variables and precaches/fx above the thread for _zombiemode::main(); and below that I call my main(); function. Look at any of my last 3 releases and you will see it is part of the install instructions.

Idk if it is proper, but I try to keep all of my script to a single script if possible. I would not thread your function in zombiemode.gsc when it notifies the betweeb_round_over.

   maps\_custom_script::init();

init()
{
      precache fx
      precache model

     level thread custom_script_main();
}

custom_script_main()
{
blah blah blah
}

That's personal preference really. I like to call all my scripts in zombiemode, that way you dont have to seperately precache everything.

I guess I'll leave my script call in mapname.gsc for now.  I'm having another issue: I followed the "UGX airstrike tutorial" to make my plane fly over the map, but when the function plane_pass_by() is called, the plane model does not move. It just disappears 

I've been into scripting recently and in a lot of things that I bumped into was this precaching. What does it do and what if you don't precache? I suppose caching is for loading things into memory so it can be accessed without "lag spikes" or whatever.

I have 0 experience with vehicle paths tbh, only time i've made a flying chopper i made animations for it in maya. But are you sure you precached everything correctly and everything is loaded in a .csv? Sounds like a missing model, or a bad vehicle-path maybe? But just guessing, like i said never used them myself before..