Leaving Huntingdon going north, the tracks are (from west to east):
Down slow - down fast - up fast - up slow.
North of the station, under the A14 there is a crossover from the down slow to down fast, but the down slow continues. About 300yds north of this is where the up slow merges with (or forks from, depending on your viewpoint) the up fast. From this point about 500 yds north of Huntingdon station it is 3 tracks (Down slow - Down Fast and Up fast) all the way to just south of Holme level crossing where the down slow merges into the down fast and it is standard 2 track from there to Fletton in Peterborough (where the spur to the Nene Valley joins) where it becomes the same 4 track layout as in Huntingdon. There is also a short section of 4 track (a freight passing loop basically) on the fens near Church End at the site of the old yard.
It is continuous 4 track from Huntingdon south all the way to Woolmer Green, south of Knebworth before the infamous Welwyn bottle-neck.
Reinstating the 4 tracks to Peterborough would be possible as in many places there is still visible track bed, but the OLE masts are often placed right where you'd need the 4-foot. There are also a few new bridges in the yaxley/south Peterborough area that would need widening. The fen stretch between Conington and Yaxley would be difficult from a construction point of view. The ground there isn't the easiest to build on. There is a 100mph limit on this stretch anyway - So like all things, it's possible if you throw enough money at it.
Benefits wise - it would help relieve congestion in the morning peak in particular around 7am to 8am when you have a lot of trains fighting for paths over the fens into London.