First, let me say, I cannot believe they had the gall to ask you to pay for the alternator. Its demise was a direct result of their negligence. Regardless of how the old battery was installed, it is their responsibility to insure that the correct cables are attached to the correct posts on the replacement battery. They goof, they pay.
There are “wires” in the engine bay that are designed to open up in an over-current situation. They are called fusible links. The car also has fuses on all of the electrical circuits to protect the wiring from burning up in the event of an over-current situation. So, the polarity reversal may have resulted in one or more fusible links and/or fuses blowing to protect the wiring. The wires themselves do not care about polarity so the likelyhood of any additional wiring harness wires being “fried” is remote.
What can happen is that any electrical device on the power bus may have been compromised. For example, your radio is likely to not only have a fuse on the general circuit feeding it and other compoenents, but also an internal fuse to protect the wires feeding just the radio. Sometimes, they do not even have fuses internally but some electrical component on the power supply front end is sacrificed so the radio becomes inoperative. If your radio is dead, it needs to be repaired or replaced (assuming the general circuit fuse is OK).
With regard to the longer cranking times, it’s probably a result of the battery power being removed from the ECM and so it lost all of the memory stores for operating parameters. It has to re-learn over time and use to custom tailor the controls to the engine condition. The older the engine, the more likely it requires some custom settings to operate best since it no longer matches the performance of a new engine. If the ECM is doing its job, it’s probably fine and in time, it will adjust to the settings necessary to operate as it did prior to the battery swap.
The rest of the electrical problems should be fixed on their dime with nothing but yes sir, yes sir, three bags full. Your problems are the direct result of their negligence and you deserve full compensation. I wouldn’t settle for anything less.