The reaction I really don't understand is the second one which produces the cycloheptatriene ring with a positive charge. The only thing I can think of is that the hydrogen is being ejected and behaving as a nucleophile but that's atypical of hydrogen. How does it happen?
Nucleophilic or hydridic hydrogen is actually commonplace in organic chemistry. Examples include:
The Cannizzaro reaction whereby a single aldehyde is reacted with base to form an alcohol and a carboxylate salt, and the variant (cross-Cannizaro reaction) whereby formaldehyde is added and it is converted to formate, the other aldehyde being reduced to the alcohol.
In both cases the hydrogen is in an environment that enriches it electronically, in the first case by the action of the base attacking the carbonyl group and in the second case by attachment to an electropositive atom. With cycloheptatriene the carbon ring is in effect also electropositive, because the cycloheptatrienyl ring becomes aromatic with a positive charge. Compare the resulting reactivity with 3-chlorocyclopropene as a source of chloride ion or, in the opposite sense of forming an aromatic anion, the acidity of cyclopentadiene.