Actually it doesn't take long for it to happen, sometimes maybe a day or two especially now with the colder nights.

 

I had that issue one time when I was using a frap air filter. Hasn't happened since I went to K&N, but we had many issues like this in a caddy alante not too long ago...

Just go to lowes or any kind of home improvement home and pick up some gutter screening. (The same chit you see ricers rolling with on their grills) Either slot a section in the side panel intake for the air box and slip in a piece of that screen, or just cut an oval to match the tube itself. It worked wonders on the alante.

 

Switch probably has nothing to do with remote start ,,, If there is no tilt switch, there should be a switch inside the vehicle to disable/enable it [assuming that it even has remote start].
If you can't find this switch, and there is no good reason to hide it real good, that is a pretty good indication there is no remote start capability.
Many vehicles will have a warning sticker on the fan shroud, warning mechanics to disable it first before working on the vehicle ,,, But of course, some are never put on, some wear off.

"Tiny button" most likely an arm/disarm button for the alarm, usually accomplished by holding it for 5 seconds with the ignition on [but not cranking].
No LED's anywhere on the dash ?

 

No LED's.

I'm going to contact the salesman, if he's still there, to see if he'll contact the original owner for me. It's been a couple of years so don't know.

I seriously don't want to have to pull that box out.

 

One option might be to take it to an alarm install shop, they should be able to take a quick look and tell you exactly what it is, and what you'd need to do to make it functional.