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.