Mow the vines??:/ Never have done that before, I usually just leave them. The runners after they have rooted can be cut off from the main plants, as they will be the future producers in years to come.
As far as the leaves, I just cut off any of the brown ones. I usually burn them or put out in the garbage, rather than compost them.
I've never cut back or pulled off strawberry leaves for any reason. Just let them grow and propagate the runners. They are about a foot tall in the summer, and "die back" to about 3 inches tall in the winter.
I have some everbearing plants so they just keep on giving.
It is a "common" practice of some to mow back strawberry mats every few years or so. This is done to rejuvenate the plants for the most part. It is done right after the stop producing fruits. It is very important that you have a long enough growing season to allow the plant to recover and come back before they are killed by a frost. It is important that the plants come back so they can work to produce enough energy to store over the winter. When you do "mow" be sure to trim right above the crown, because if you damage the crown the plant will not comeback and therefore the plant is lost.
So, in short, to answer your question, you "mow" right after they stop producing fruits and only do it every few years or so if the plants become thick and matted.
Mowing, IMHO, is not necessary if the plants are kept well thinned from runners and you trim back the dead leaves each year. I can see where this would be useful if you have a huge strawberry patch that would be seemingly impossible to tend to each plant individually.
That makes sense if you have a huge patch and don't have the time to dig up and relocate plants every 3-4 years. However, digging up and moving a few hundred plants isn't that bad if you have the time and space to have the new beds pre prepared before the great dig up.
Yes, that's what the strawberry farms here do. They transplant nicely.
Another method is to have 2 rows, one behind or in front of the one the strawberries are in. The runners tip down in the next row, and voila, instant strawberry row.