@wantafastz28
The device with the batteries in it that was mounted outside the alarm panel is the Wired Zone to Wireless transmitter. Basically this takes the info from the wired zones and sends it to the Lynx Panel wirelessly.
The GSM dialer with Sim Card is inside the Lynx Panel. If you open the main control panel it is mounted inside with a single screw and the sim card is connected their, it just has a basic bridge/connector that connects and powers it off the main panel.
As for the garage sensor - Honeywell makes a wireless Tilt Sensor - I've only used it once and I never had an issue with it. Be aware though you'll have to factor this into your setup. For example the garage can't be open when you go to set the alarm as it will show a fault (you can bypass it and set it, but that is just more steps) Also, when you arrive home and the door goes up the alarm trip will occur, so you'll have to factor that into your wait/alarm time. For example most people set entry to like 10 or 15 seconds meaning you have 10 seconds before alarm sounds when first triggered. I don't know about you but by the time the door goes all the way up, i get out, get the kids out, and head inside I'd need like a 2 min delay, which totally defeats the purpose because if someone kicks in the front door they'll have 2 free minutes... This is why I dislike the garage sensors. HOWEVER, you set them as passive so you simply use them to view the status of the door (fault is open, closed is closed), or you could make that zone only active when armed "stay", so its only enabled at night while asleep when everyone is home... Just a few ideas.
The other possibility since you have left before with the door open is to look into something like the Chamberland MyQ wireless garage control system which will allow you to see the status of your garage door (open/closed) as well as open or close it remotely so if you did leave it open, you can close it via your smart phone. While an alarm sensor will only tell you if it's open or closed, and still require you to return home to correct the problem.
The main alarm box is an older hardwired system control board with limited features/power support. The wires that still are connected are the battery and most likely the wiring for the old control panel (possibly had 2? Hard to tell without a closer pic). Also the fact that a heat sync and larger processor is missing suggests it being an older model - most new controller boards have large heat syncs to handle larger power loads/draws for numerous zones/key pads as well as a larger processor to handle data.
As for opening up the Lynx panel - their won't be much to see honestly. You'r going to have your main power cable that goes to the main board and then your cell dialer, and thats about it. Really isn't anything to adjust or mess with inside that self contained unit.
Bonus to all this is all your wired sensors are still intact at the main box. You already have power at the box to power a panel/battery if you put in a new system. So once your contract is up if you want to order a 20P system off eBay you can for $75 to $200 depending on what all you want with it in way of additional keypads/sensors etc. At which time you can look at a product called Eyezon if you want to self monitor with remote/mobile control with no monthly fee's.
EyezOn | Internet Enabled Home And Business Security Alarms For Smart Phones or if you want monitored you can look at something like alarm relay or alarm grid depending on the features/support you want.
Good Luck! At the very least you are advancing your knowledge around the technology that is already in your home!