Its not true. But walking gives an easier labor because you are more healthy, it also helps pump more oxygen to your baby.which is amazing.... Swelling is normal, most pregnant woman get it. No need to worry. Unless it's extremely painful, then id worry....Though u can look it up on google to help ease your mind.
I worked out and walked from day 1. I had to have a c section because he wasnt head down and the stupid drs didnt find that out until middle of week 39 and kept saying he was in the correct position until i went into the hospital and they checked my fluid level by ultrasound and found that out and i had to have an ultrasound the next morning because they said it was too late and he was too big to turn. I was so disappointed, but its better than laboring and then having to have an emergency c section i guess.
This is just my opinions, should not be taken as medical advice :) I'm not sure about walking help to avoid a c section but it should help induce labor and any exercise is considered helpful when she does go into labor. For the swelling, if it's a lot of seeking on one side, she may have a blood clot (my Dr was concerned about the same thing with me and u had to get an ultrasound on my legs to rule it out) but it could just be normal. Wouldn't hurt to ask the Dr anyway. Hope this helps. Good luck and congratulations!!!