My Buick used to do the exact same thing. First bought the car, figured out it would only start after 5 sometimes 10 minutes, then start like there was nothing wrong with it.
Alarm had some kind of feature that cut power to the starter wire, or power to the starter, key would act like there was no battery in the car until enabled, then juice would be sent that's power ok and car would start.. this one if those alarms, all digital. You could turn car on, have it running, lock it, beeps three times all doors would lock, beeps twice only driver door would open, beeps once was proximity alarm, tap teice would start the car, tap once would unlock doors, then an alarm button.. so complicated I could not even figure it out. Even had a Carjack mode ? where digital keyfob would display a picture of a gun .
Paid $300 for that car, pretty sure it was a 98 Regal. Sold it for $1000 to someone who assured me fixing it would not be an issue (we agreed it was likely an alarm install that gave the starter wire issues and they probably removed it.) Alarm retailed for about $400. And no it was not a Viper alarm, I don't know what it was.
The short answer is.. yes, it can cause issues. And I am not sure I would ever want an aftermarket alarm on a car, ever. Ever.
If it has a keyfob and can unlock the doors and maybe blip the lights, that's all you need. Test it.. leave the hood open, lock the doors, get out, close hood. Alarm should blare and turn off when you hit Unlock key. And if it gets stuck on, as I've had happen.. locking and then unlocking driver door with the key will shut it off too.