Here is possible implementation that follows all constraints:

```ts
interface Pair { start: number; end: number };

export function mergeIntervals(pairsList: Array<Array<number>>): Array<Array<number>> { 
  // Converting list of pairs to a list of pair objects.
  const intervals = pairsList.map(([start, end]) => ({ start, end }));
    
  if (intervals.length === 0) return [];

  // Sorting pairs by their starts.
  intervals.sort((pair1: Pair, pair2: Pair): number => {
      return pair1.start - pair2.start;
  }); 

  const mergedPairs = [ intervals[0] ];
  for (let i = 1; i < intervals.length; ++i) {
    // If the current interval overlaps or touches the previous one, 
    // merge them by updating the end of the previous pair:
    if(intervals[i].start <= mergedPairs[mergedPairs.length - 1].end) {
      mergedPairs[mergedPairs.length - 1].end = Math.max(mergedPairs[mergedPairs.length - 1].end, intervals[i].end);
    } else { // Else if the current interval does not overlap with the previous one append it to merged_pairs:
      mergedPairs.push(intervals[i]);  
    };
  // Throw error if a pair's start is greater than its end
  if(mergedPairs[mergedPairs.length - 1].start > mergedPairs[mergedPairs.length - 1].end) {
      throw new Error('A pair has an invalid range');
    };
  }

  // Converting pair objects back to their original format.
  return mergedPairs.map(pair => [ pair.start, pair.end ]);
}
```
This implementation will handle edge cases correctly like single or empty input and it ensures that all the pairs meet the constraints of a valid pair: start should not be greater than end. It maintains sorted order so it also fulfils the requirement for output to be sorted by interval start.