As for value if you see that the landscape is less soggy, then it is doing something. I would have gone to a more standard functional design as such 3" to 4" pipe that is a minimum of 12" down, backfilled with gravel, and warped and covered with filter cloth. If you are going to take the energy to do something, down to capacity and correctly.
1. Trench Part A...would it be better for me to remove the pipe? Does it really add value?
I would leave as gravel.
2. Trench Part A Would it be an issue for me to cover the top with fabric, and bury it in soil and sod it or should I leave it as a gravel path?
No actually soil on top would not give you that option because you do not want to bed concrete or stone in soil but on the gravel. Only undisturbed mineral soils that naturally drain well, that are below a given forest line are ever suitable for setting stone or concrete.
Reason I ask it is soil on top would give me the option to put a concrete sidewalk, pathway or steps there later.
3. Trench Part B...Would it be better for me to remove the pipe from my trench and just use rocks since it is so shallow?
3" or 4" pipe is standard and would have still been best in your case. As for slope you would have created that in the bottom of your trench. 1/4 for ~ every 10' horizontal travel would have been sufficient. 2" has tendency to "silt in" even with cloth.
This is great to know! I appreciate all this info. I used 2" pipe rather than 4" on both A and B because Trench B does not have the slope to use 4". I used the largest size that would fit both. You're right I do want to do it correctly but also make the best of a limited slope.
If you are going to redo it this will be fine for now.
Would it work if I make both without pipes and just have an open gravel bed that feeds a buried one?
(wrapped in landscape fabric) Would it be a good enough solution for now
Next year I plan to revamp the whole thing.
Depends...water goes to gravity and path of least resistance.
If there is any slope then water should prefer a graveled path correct?