Chris, your last 2 examples would be said ‘ J’ai laissé tomber qqc … ‘ or ‘ il a laissé tomber qqc‘ ( laisser tomber qqc ) not ‘J’ai or il a tombé .. qqc ‘

 https://www.wordreference.com/fren/laisser%20tomber

There are limited expressions, mostly informal, some even lower register than ‘informal’ (beware),  in which tomber takes a direct object and is conjugated with avoir in compound tenses. 

I don’t think these limited exceptions warrant a separate lesson, although perhaps a brief note that they exist may be reasonable. 

 https://www.lawlessfrench.com/expressions/tomber-expressions/ https://www.larousse.fr/dictionnaires/francais/tomber/78343

The verb tomber uses the auxiliary verb être in the passé composé when it is used intransitively, indicating a change of state or movement. However, when tomber is used transitively with a direct object, it uses avoir. As a general rule: every transitive verb, i.e., one that has a direct object, uses avoir as an auxiliary.

 Here are examples of both cases:

Intransitive (with "être" as auxiliary):

Les feuilles sont tombées des arbres. (The leaves fell from the trees.

Transitive (with "avoir" as auxiliary and "laisser tomber" as the more idiomatic usage of "to drop"):

J'ai laissé tomber la boîte par terre. -- I dropped the box on the ground. (la boite is COD)
Il a laissé tomber son stylo. -- He dropped his pen. (son stylo is COD)

@Maarten: Yes, you're right. I edited my answer slightly.