Hello,
np, let me try.
For the first one - string sorting, we can start with creating a wrapper class to store data.

Public Class DataWrapper{
String NPI;
String fromDate;
String toDate;
}

Store all data in a Map with NPI as key.
Map<String, DataWrapper> dataMap = new Map<String,DataWrapper>();
Sample Map Data: {'01:1245678' => new DataWrapper()}

List<String> compositeKeyList = new List<String>();      //Contains data which will be in sortable format.
Composite key for each DataWrapper would be like :  07/25/2020-01:1245678     (Remove 01 from fromDate)
add all dataMap DataWrapper composite key to compositeKeyList by looping through.
After adding some data list would look like : {'07/25/2020-01:1245678','06/29/2020-02:1245678'','07/16/2020-03:1245678'}
compositeKeyList.sort();           //Sort the list

Loop through the compositeKeyList and arrange the DataWrapper data for use :
List<DataWrapper> sortedData = new List<DataWrapper>();                    //Final data which is sorted by fromDate

for(String v_CompKey : compositeKeyList){
  String v_NPI = v_CompKey.split('-')[1];                //get 2nd value, which is NPI
 sortedData.add(dataMap.get(v_NPI ));                 //get data from dataMap using NPI and add it to list
}

Try and see.

If it doesn't work bubble sorting algorithm (https://www.geeksforgeeks.org/bubble-sort/) can be implmented on DataWrapper, it will only change the implementation to use custom method to sort instead of using sort(); 

Thanks