Thermactor (air pump) puts air into the exhaust. Doesn't change how the engine runs. The exhaust still being hot enough to burn, but just lacks the extra oxygen, does burn itself cleaner. Dirt simple emissions control device. You can spin the air pump by hand, it doesn't take any noticeable power to run.
EGR is only a part throttle cruise application. At full throttle it is closed and does not affect power. At cruise it increases cylinder pressure without adding any extra fuel. Due to the inert properties of already burned exhaust it helps control NOx emissions and the added cylinder pressure improves fuel economy. The bad part is the tuning on carbureted applications can be problematic. How much flow, at what times, just isn't the best at times and has a reputation for drivability issues. If it works good hooked up it is worth keeping hooked up. But working poorly it can cause issues. More issues can arise by unhooking it as the carburetor should be tuned for the EGR to be functional at part throttle. Removing the EGR can cause part throttle drivability issues. Full throttle the EGR has no vacuum and is closed, just like it isn't there.
Cats are great at cleaning up exhaust. Older carbureted applications almost always have an air pump as part of the operation. They burn off the un-burnt parts in the exhaust. They need enough free oxygen to do so. EFI engines with O2 sensors have enough feedback that the engine can pump enough oxygen into the cats to keep them working. On a carbureted engine the tune still has to be pretty close. Try to clean too much dirty exhaust and the cats can overheat and melt down.
Old school pellet bed cats are a noose in the exhaust system. Modern honeycomb cats flow really nice. Todays aftermarket high flow versions don't have negative impacts to flow. Most of the "more power" claims by removing them are from the added noise you get.