Problems with 7.gen Celica

[T6nnS]

My family bought in March a 6 year old Celica. It's not the T-sport version, just the normal 1.8l with 105kw. Anyway there are some problems with it and any advice is greatly appreciated.

First problem is that the engine revs flactuate when at idle (no gear in). It flactuates from 600-1500. And maybe the other problem might be related to this, the problem is as followed: when in gear and pressing the clutch the revs rise about 500rpm and then it drops to idle speeds.
Theres an issue with the factory alarm. When it's turned on for a little while, everything is fine (alarm doesn't go off), but for longer periods (like for a night) when taking off the alarm the alarm goes off (sirens starts and lights flashing).

As you might notice my english isn't very good and I hope someone can make something out of this and help me

[Blane]

Let's give a new member some help. I'm not familiar with these problems. alarm is too sensitive is a given.