If AA causes you to miss your connections they will rebook you on the next available flight at no charge. Whether or not they pay for a hotel room is dependent on the cause for delay primarily with other factors such as fare paid and status also causing exceptions. For example, your chances of having a hotel paid for in cases of weather are almost zero. However, if a mechanical happens resulting in the cancellation, the odds are almost certain that you would have your lodging taken care of.

In your 2nd scenario, if it's all on one ticket (one PNR), then the same rules apply as far as being accommodated. However, TAM is responsible for rebooking you.

Although Chris Elliott is an idiot and generally a poor reporter, there is a degree of truth to the article. Your questions don't relate to what happened to the subject of the article. Many of us on FT would have replied to the subject that she should have taken matters into her own hand and flagged down a supervisor at the 60-90 minute mark and informed them that she was getting tight on time... and kept bothering them as time got closer and closer. The additional fare of $2600 is fishy and isn't consistent with the rules as I know them.