No, you do not have to worry about new popcorn ceilings having asbestos. All residential building materials used since the mid-1980s stopped including asbestos, so new popcorn ceilings have no risk of asbestos. You can still create a textured drywall ceiling with a popcorn texture if you like, but new techniques have no risk of adding asbestos. 

No, not all popcorn ceilings have asbestos, but a vast majority do. The Environmental Protection Agency (EPA) banned its use in 1973, although it still may have been used up through the early ‘90s. But, in general, homes built or remodeled before the early ‘80s likely utilized materials with asbestos. That being said, a professional test is the only way to determine for sure.

No, but it’s common that they do. Gravity furnaces weren’t terribly efficient so wrapping either the ductwork in asbestos, the furnace itself, or both, was something done regularly. This allowed the system to retain more heat. At the time, this greatly enhanced the efficiency of the furnaces but it wasn’t until later that we became aware of just how carcinogenic and dangerous asbestos could be.

If you have an older electrical system with fraying wires and other signs of damage, removal  may be the best course of action. When asbestos begins to break down, it's more likely to become airborne, increasing the risk of health issues when it's inhaled.

However, if the system is in good condition and doesn't have visible signs of wear or damage, leaving it alone may be the safest course of action. An asbestos abatement and removal professional can help you determine the best way to proceed.