I'll try and answer most of your questions:
Q: Can I send SMS messages to the device to control its functions? I don't need anything else other than simple messages to control the board. The simpler the better.
A: Not sure, I haven't used SMS myself. I've been working with UDP messages.
Q: Will I have to do anything special to make SIM cards work around the world or do I just plug them in and they start working?
A: You do need to set some settings depending on the SIM provider you have. APN names and such.
Q: When the board is idle, i.e. connected to the network but not transmitting or doing anything else, what it the power consumption? Low (average) power is very important to me.
A: The board draws a decent ammount of power in IDLE mode, luckily the board has a Power Save mode in which it draws 3-4uA in which it can remain connected to the network.
If power ussage is important to you, I suggest you look into using NB-IoT instead of LTE or GSM. NB-IoT is way more power efficient than both of them. The R410 has some nice features like early release that make it so that it only draws about 150mAs. See this forum: https://forum.iot.t-mobile.nl/topic/278/how-much-battery-lifetime-can-we-expect-with-a-sara-n200-module-on-our-iot-network/2