Least amount of ender pearls and blaze rods to leave the nether? (speedrun)

Solution 1:

Least is zero, in Set Seed runs on a seed where all end portal sockets are already settled.

On 'random seed glitchless' which is currently about the most popular category, 13 pearls is a safe amount. Get less and you're risking.

First off - regarding the end portal:

The frame has the highest probability of generating with only one eye of ender (37.7%), with probabilities dropping to 28.2% for zero, 23.0% for two, 8.52% for three, 2.13% for four, and a total of 0.433% for five or more.

Next, eyes of ender:

The eye has a 20% chance of shattering. The player can throw the eye an average of five times before it shatters.

Finally, stronghold loot:

In Java Edition, each stronghold altar chest contains 2–3 item stacks, with the following distribution:

Item          [...] Avg. # chests to search  
Ender Pearl   [...] 4.3

So you need 11 pearls to be mostly on the safe side regarding the portal - only 28.2% of strongholds don't have one. You need more to find the stronghold - it's unlikely not to break at least one eye of ender in the process, but that part is in equal parts luck and your own skill at triangulation - a great player will get the stronghold in 5-7 throws, a beginner may require 20 and more. And last but not least, if you're one pearl short, the run isn't dead yet, as chances are good there's one in a chest in the stronghold.

Summing up, with 11 pearls you need great luck - not break any, and have at least one in the portal, or if you break one - have two in the portal. With 12, you still have a good deal of a chance of losing - if one pearl breaks and the portal is unsettled, or two break. With 13 and more, you're mostly safe.

Obviously 1 blaze rod produces 2 blaze powder, so you need at least 6, preferably 7. If you have more pearls than 12, don't craft the last eye of ender unless you actually see you need it to fill a slot - a spare enderpearl can save a bunch of time getting from the spawn platform to the central end island in case you spawn way off the edge of it.

edit:

Regarding number of drops, I ran this rig for a good while, used up 12 stacks of gold, produced over 3 chests of 'junk' and obtained 258 pearls, in the following ratios:

batch size    # of batches
4             10
5             8
6             6
7             10
8             9

It's still a bit too little to have full statistical confidence but it seems there is no special bias of any kind and all the allowed trade sizes appear at roughly the same frequency.

This applies to 1.16.1 - there's no point speedrunning 1.16.2 or higher, as there's no separate category for these and both pearl trade frequency is halved, and the trade amounts are 1 to 4 instead of 4 to 8 pearls.