What is the rationale behind forcing passengers to check in dangerous items?
What makes liquids and gels more dangerous in the cabin? Why would a lithium ion battery be less dangerous as check-in luggage? Wouldn't that even be more dangerous when you put lots of batteries in the cabin, since one could catch fire and spread to all other batteries where passengers are?
Generally, "dangerous" things that might be used as weapons are required to be kept out of the cabin. For instance, you can have firearms in checked baggage, but not carried in the cabin.
Given what's been learned about terrorist planning, there were plots to bring liquids on board an aircraft that could be combined to form an explosive. To counter that threat, the amounts & types of liquids that can be brought on board have been limited. Since the liquids in isolation are safe, they're allowed in normal quantities in checked luggage.
In the case of Lithium Ion batteries, the concern is less their use as a weapon, and more about a defective battery catching fire. If this were to happen in a cargo hold, the fire would cause other things to start burning as well before the cargo smoke detectors alert the crew to what's going on. So the priority is to keep lithium ion batteries out of the cargo hold. They're allowed in the cabin (with some restrictions) since you're pretty likely to notice your cell phone getting hot & smoking if it's in your pocket, and with that sort of early notice, the fire can be dealt with before it's caught large volumes of material on fire as well.
At least that's the theory. I don't know how many times it's been put to the test. In at least two cases (the Samsung Note 7 phone, and hover boards), devices which are considered to be at high risk of catching fire are simply kept off the aircraft altogether.
Why is the rationale of forcing passengers to check-in dangerous items?
To stop people using them as weapons to hijack the plane.
What makes liquids and gels more dangerous in the cabin?
There was a terrorist plot in which suicide bombers would blow up planes using liquid explosives disguised as soft drinks. Of course, somebody could try to put such a bomb in checked luggage but that would require some sort of timer and detonator, which would likely be spotted when the luggage is X-rayed. So the specific concern is that a bomb could be assembled and detonated in the cabin using only innocent-looking liquids and other items.
Why would a li-ion battery be less dangerous as check-in luggage? Wouldn't that even be more dangerous when you put lots of batteries in the cabin, since one could catch fire and spread to all other batteries where passengers are?
Lithium batteries are more dangerous in check-in luggage.
How would the fire spread between batteries in the passenger compartment, unless the whole cabin is on fire? Your laptop isn't suddenly going to start burning just because my cell phone has caught fire in the seat next to you.
If a battery fire starts in the cabin, somebody will notice before it sets much else on fire. They can then put the fire out. The standard technique, I believe, is to drop it in a bucket of water, which removes all the heat from the system, as well as starving it of oxygen. Everybody lives to tell the tale.
In contrast, if a battery fire starts in the cargo hold, it's in an area that's densely packed with flammable things such as clothing. That means that, by the time the fire is detected, it is probably quite big. Further, the only way to fight such a fire is with a general fire suppression system that tries to starve the entire hold of oxygen. That doesn't remove the heat so, even if it puts the fire out, it's possible that the still overheated battery will reignite the fire. This is an extremely dangerous situation that could well kill everybody.
