Craig,
Most your problems will go away when your engine warms up. That's how engines are designed. They could be correct when they say your hard downshifts are results of your backfiring engine. Though, early '90s Diamantes have been known to have transmission problems. But since yours was replaced, we'll look at the backfiring problem.

A backfiring engine can have many causes. It is possible for a clogged catalytic converter to cause a dramatic drop in engine performance. It is also feasible for excessive exhaust system back pressure to cause the engine to backfire.

Sometimes the backfiring blows the exhaust system right off a vehicle. Then, engine performance returns. Also, there are other causes for engine backfires of this type.

The backfire described is not the type that blows out the exhaust system. It appears this is a backfire in the intake manifold of the engine. This could be caused by faulty spark plugs or spark plug wires. Also, an excessively worn timing belt, valve or camshaft could be the cause. Perhaps a valve lifter has failed or a valve spring has become weak from age.

Before spending a lot of money on exhaust system components, it would be a good idea to ask an experienced technician to determine the cause of the backfiring. If internal engine wear is the cause of the trouble, it may cost more to repair the engine than the vehicle is worth. Keep that in mind, but don't give up on the car all together. The Diamantes are great cars and can last a long time. Yours is pushing 11 years, and that's about middle age for that car if maintained well.

Good luck!