Is legal to send my GSM IoT device which is powered by a ~2000 mAh (3.7 V) lithium polymer (LiPo) battery via air post? (Or even take it with me inside my suitcase?)
Also does it make a difference if the device is powered on or off?
The IATA Lithium Battery Guidance Document is as close as you can get to a worldwide set of guidelines for battery transport by plane.
The section which I assume applies to your device is 126.96.36.199, which states:
(a) each installed or spare battery must not exceed:
for lithium metal or lithium alloy batteries, a lithium content of not more than 2 g; or
for lithium ion batteries, a watt-hour rating of not more than 100 Wh.
(b) batteries and cells must be of a type that meets the requirements of the UN Manual of Tests and Criteria, Part III, subsection 38.3;
(c) articles containing lithium metal or lithium ion cells or batteries, the primary purpose of which is to provide power to another device, are permitted in carry-on baggage only. These articles must be individually protected to prevent short circuits by placement in the original retail packaging or by otherwise insulating terminals, e.g. by taping over exposed terminals or placing each battery in a separate plastic bag or protective pouch.
(d) if devices are carried in checked baggage the passenger/crew member must take measures to prevent unintentional activation
Your battery has approximately 7.4 Wh of energy (2000 mAh * (3.7 V / 1000)), which is very low and unlikely to be considered a powerful battery, and is well below the given limit.
However, the best advice would be to consult with your airline to ensure that they do not have more strict rules than this (and don't wait until you get to the gate - ask well in advance because it's likely that most staff will not know the exact rules).
As for the GSM chip, you will need to fully disable any hardware capable of electromagnetic interference (which would include this chip). If it will be stored somewhere that you cannot access throughout the flight, the best course of action would be to remove any sources of power completely (i.e. the battery).
If the device is powered, and the radio is capable of being powered, then this is likely to be prohibited by the carrier. You should find the specific carrier will provide regulations for what can legally be shipped on a specific route. The final consideration is that encryption technology is also restricted for international shipping.
As a GSM capable device you must switch it to Airplane mode. Which means the device RF interface must be completely shut down.
Basically what you have described can be any smart phone, and I believe those can be delivered by air post as long as they are in airplane mode. But certainly if they are powered off.