Are you willing to say what you’re trying to accomplish with this project? What are you trying to monitor? To what end? Is this a product you want to commercialize and sell or something you just need to solve a problem you have?
Essentially the device will be powered off a 12 Volt battery in remote areas without a 120V power source in some cases.
Will there be solar panels or something charging the batteries or do they have to last a long time on their own?
(in some cases people may put this appliance near their home but that's the easier case).
It might make sense cost-wise to use a different cellular communicator for the indoor vs outdoor/industrial case.
1) What component would you recommend to act as my "alarm system" that I could buy off the shelf?
An easy off-the-shelf way to detect voltage drop is to use a relay module like the Altronix RBSNTTL or something similar.
http://www.altronix.com/products/installation_instructions/RBSNTTL.pdf
There are many relay variations. If you can find a physical relay module with the right input trigger voltage and a low enough current draw then you don’t have to do electronic circuit design. Otherwise, you can design a transistor switch with an open collector output.
(keep in mind this will be in an environment that may get down to -10 F in some installations but will be dry)
This would probably be more of a factor in the cellular communicator than the detection circuit.
2) I would need to setup communications with a company such as Verizon for 3g or LTE alarm service... any recommendations where to start contacting them with my device or is there a standard Alarm device that I bolt on that I could use (this is similar to my question 1 and may already be answered)?
You haven’t said much about your application so I’m making some assumptions but based on what you have said it seems easiest stick with an off-the-shelf cellular alarm communicator that has a wired sensor/contact input. These things are usually made to be wired to a security alarm panel but many of them have 1 or 2 sensor inputs built in for monitoring stuff in a dedicated fashion like this, rather than using with a full security alarm system.
If you stick with a cellular alarm communicator that sends Contact ID or SIA signals then it would be able to communicate with a central monitoring station or send you text messages, etc…
One example might be the IP Datatel IPD-CDMA-L which says it can operate down to -22 degrees Fahrenheit and has 2 zone inputs that could be wired to a relay contact output.
https://ipdatatel.com/cellularcat.php
https://ipdatatel.com/pdfs/Cut%20Sheet%20CAT-L.pdf
Another example might be the Videofied XTO-IP630 which is a full alarm control panel and would cost more but it also does more. It operates down to -20 degrees and has 3 wired sensor inputs. Videofied is really good at battery life conservation. According to the datasheet the control panel can run on just D batteries without charging for up to 4 years.
https://www.videofied.com/_asset/0mtjcb/Outdoor-Panel-XTO-IP-3G-Specifciation-Sheet.pdf
I’m sure there are other cellular communicators that work in the cold but those came to mind off the top of my head.
3) How much energy draw does the above recommended device pull? I'm envisioning using a 12 volt battery (use this as my basic capacity) to power my appliance invention and this alarm.
That depends on what parts you choose, of course, but you can look that up in the datasheets.
My opinion might change if you share more information and my assumptions are wrong but I think you may be able to accomplish what you need with a simple relay module wired to cellular alarm communicator and use some kind of cellular alarm service like IP Datatel, Uplink, Telguard, etc…
I wish I could recommend an Alarm.com system for this so you could use our DIY service but there are no Alarm.com compatible control panels or cellular communicators that would work stand-alone or in the cold weather. You need something more industrial.