I batch sparge, and I've tried most variations on the idea over the last 13 years or so (from no-sparge, to 4 batches). I've tried various thickness mashes from less than 1 qt/lbs to almost 3 qt/lbs. I've chased efficiency up into the higher 80's. Here's some of my thoughts:
Process changes: Since you are new, I recommend that you don't change anything for several more batches. Continue hitting your numbers for 5 or so more batches. This will give you the confidence to KNOW that you have a working, consistent process. From there you can know that any variation that you see AFTER YOU MAKE A PROCESS CHANGE is caused by the change, and not simply due to inexperience.
Number of batches: a single batch is sufficient. Switching to double batch will get you another 5 percent efficiency, and cost you another 15 minutes. Not worth it in my experience. The only thing I do towards "maximizing efficiency" is to ensure that my two runnings are approximately equal in volume (3.75 gallons@room temp, each).
Efficiency: The higher you go in expected efficiency the more variation you will see from batch to batch. Ie, different minor variations in ingredients, grain crush, mash pH, temperature, mash duration, etc begin to result in changes in real extract. One session might get 85%, another 82%, another 87% (and the oddball 65% for no good reason). I find nothing more frustrating than UNDERshooting my pre-boil OG. If my recipe calls for 1.054 and I only get 1.047, I'm NOT HAPPY. Setting my efficiency expectations lower (70%) allows me to MANAGE the process and ensure that I get the desired results with less variation from session to session. I pay a minor additional cost in grain to do this (generally about 1-2 lbs of extra grain per recipe).
Final Runnings Gravity: Generally, based on my target efficiency of 70% my final runnings end at around 1.020. This varies based on target OG, of course, but for your basic 1050 beer, 1020 is where I end. A low 1040 beer would be in the low teens (101x). I don't ever go below 1010, no matter what---add water to dilute lower than that.
Temperatures: Mash out temps are not necessary. Most of us do it, but there's no real reason to (it doesn't increase efficiency, it doesn't really denature the enzymes instantly to halt conversion, etc). Your 170F water and 155 sparge bed temp is just fine. If you want to mash-out anyway, 190F water is the highest I would recommend with a plastic cooler. Anywhere between 170F and 190F water temp is fine. Just pick a temp and be consistent. Others have succeeded at 200+, YMMV.
Stirring: stir at dough-in, just as you did. The goal is to ensure that you have no clumps of dry grain/flour (dough-balls) in the mash and ensure that every bit of grain is wet. Once that's done, stop. Don't stir during the sach-rest. It doesn't help anything, and it just drops the mash temperature by 2-3 degrees each time. DO stir in any sparge addition. Stir vigorously for 30 seconds or so. Waiting after stirring in the sparge is not necessary.
In summary:
0. Don't change anything at all for several more batches (including any changes recommended by this post). Be sure you have a consistent process.
1. Two sparges, of equal runnings. Each 1/2 of your total pre-boil volume.
2. Target an efficiency around 70% is fine (that will generally result in final runnings of 1.020 or so).
3. Mash out is not necessary. Do it, if it makes you feel better...but, it isn't doing anything.
4. Stir whenever you make an addition, but not otherwise.
5. No delay after adding sparge water is necessary before draining the MLT.
My "simplified" sparge process looks like this:
1. Add strike water to MLT (grain absorbtion + 1/2 pre-boil volume).
2. Stir in grain.
3. Wait for sach-rest to complete.
4. Recirculate wort until no grain material comes out.
5. Drain first runnings into kettle, and fire burner.
6. Add 190F sparge water to grain.
7. Stir to break up the grain bed and even out grain/water mixture.
8. Recirculate wort until no grain material comes out.
9. Drain into kettle.
10. Continue to heat to a boil. Take pre-boil measurements after you reach a boil.
I'm a huge proponent of measuring. But, I don't want to overwhelm you with information. Do the above for several batches. Get used to the process and enjoy making some beer.