Answers
If a pilot says that he/she is "working" under a particular section of the FARs, it means that section governs his/her operations. For example, Part 135 governs charter and air tax ops (some commuter airlines ops have fallen under 135 as well). Part 121 govern scheduled airline ops.
Multiple sections of the FARs govern most operations. For example, at Purdue you might be flying at an FAA-approved flight school. Part 141 governs such schools. However, some aspects of Part 61, which govern airman certification, may apply. All flight operations fall under Part 91, which set forth the general operating rules for all flying.
Your Commercial certificate only means you can fly for hire. The FARs may affect the particular flight operations for which you are working, and the minimum time requirements. For example, you need 500 hours to be PIC under VFR for Part 135. 1200 hours and a few other things for IFR PIC under Part 135.
You are correct that you must possess an Airline Transport Pilot Certificate to be Captain at a scheduled airline. Accordingly, you must have at least 1500 total hours to be eligible for an ATP.
As far as the regionals are concerned, each airline sets its own minimum requirements for hire. When I was in the regional airline market 10 to 15 years ago, the regionals' standard minimums were 1500 total time and 500 of multi. Some demanded the ATP prior to hire; for others, you needed an ATP to warrant any serious consideration. There are some regionals who hire FOs with less time; i.e. Mesa has been 1000 total-100 multi forever. Quite a few regionals now have lesser quals than the 1500 total-500 multi I stated above.
As a practical matter, there is a big difference between published hiring minima and competitive minima. The quality of the applicant pool drives competitive minima. Therefore, you may need substantially more experience than published requirements to receive serious consideration. On the other hand, when hiring is brisk, airlines may not be able to find applicants who meet their quals (do not construe this to mean "pilot shortage" - there is no pilot "shortage"!). In that case, they lower their quals and the void is filled.
Hope that helps.