Find jobs and dependencies run between two timestamps

pgAgent does not require a graphical user interface at all. It is a process that runs as daemon on unix. pgAdmin is used for configuring the job details. The configuration is stored in database postgres where pgAgent will read it. Once you configured your jobs, you may disconnect pgAdmin safely.

Alert log is a database log file where you can find various messages about your database, like time of startup, some db parameters, log file switch, possible errors that can occur etc.

Archive logs are archived redo(online) log files. In redo log files are all changes that happened to your data. If your database is in archivelog mode, than redo logs can't be overwritten, they are archived in some other location when they are full. Redo and archive logs are used in various database recovery scenarios.

So your Oracle expert probably used LogMiner(or some similar tool) to search through archive log files to find something like truncate table some_table;