Alum does not improve the firmness of quick-process pickles but may be safely used to firm fermented pickles, although it is unnecessary.

The calcium in lime (mineral) definitely improves quick pickle firmness. Food-grade lime may be used as a limewater solution fo r soaking fresh cucumbers 12 to 24 hours before pickling them. Excess lime absorbed by the cucumbers must be removed to make safe pickles. 

How to Remove Excess Lime

Drain the lime-water solution, rinse, and then resoak the cucumbers in fresh water for 1 hour. Repeat the rinsing and soaking steps two more times. To further improve pickle firmness, process cucumber pickles for 30 minutes in water at 180°F. This process also prevents spoilage, but the water temperature should not fall below 180°F. Use a candy or jelly thermometer to check the water temperature. 
Source: So Easy to Preserve, NCHFP

 

Grape leaves contain tannins that inhibit the pectinase enzyme that can make pickles soft. However, removing cucumbers' blossom ends, which are the source of undesirable enzymes, makes the addition of grape leaves unnecessary. Most modern recipes do not use grape leaves, which can sometimes impart a bitter flavor.

An ice bath firms up the cell wall of the cucumbers and might help crisp refrigerator pickles, although it is not necessary. Keeping pickles in the refrigerator will also result in a crisp pickle.

You may reuse commercially-bought pickle brine to flavor pickled vegetables but not to preserve them. The acid-to-water ratio is not the same as a fresh brine but will impart flavor to whatever vegetables you add to the jar. Let stand for 2 to 3 days in the refrigerator before using, and use within 7 to 10 days. If you observe signs of spoilage, such as bubbles, odor, or changes in color, then discard.

 

No, use pickling or canning salt. Most other salts have additives that would compromise the fermenting or canning brine solution.