Q: Are pingbacks sent immediately when a post is published, or are they scheduled as a cron job? If the later is correct, how often does the job run and can I trigger it manually?
A: You can install core control (wordpress plugin) to find out more.
Q: Are there any other terms for PBs to be sent besides having “Attempt to notify any blogs linked to from the article” set to on, and of course a link to another blog in the post content? (for example, should the sender post be public and not private? should the blog be non private?)
A: WordPress can and does not care about external resources being somewhat public or private as somewhat undefined terms.
Q: Should the outbound link be placed in the post content or can it be placed in a custom field of the post in order for the PB to be sent?
It need to be placed into the content.
Q: In case PBs are not sent nor received in my blog, what is the best way to debug and detect the problem?
Log and analyze the in- and outbound network traffic of your site with a focus on HTTP requests to find out more what is going at all and probably going wrong. Additionally, the core control plugin I linked above can be helpful for cron related debugging.