Older strawberry plants generally die in the second or sometimes third year after they flower and put out runners. Not a big deal, the runners will continue on. You can let the patch grow to any size you want -- a few square feet or hundreds of square feet, it's up to you. When the patch is the size you want, you can either turn the runners on the edge back into the patch before they root, cut them off before they grow in to the lawn, or just let them root into the lawn and let the mower cut them off.

First step is correct the heart of the problem by getting the shade off the area, correct you acidic soil PH, and provide proper drainage. Any broad leaf herbicide will kill it, but unless you correct the conditions, they will just return.

Hard to say. They spread by runners and if its anything like creeping charlie its hard to know where its about to come up. If you want to go non toxic try a product like Burn Out II which is vinegar, lemon juice and clove oil or an herbicidal soap. Both work by degrading the leaf surface, albeit from different ends of the pH scale. Doubtful that one app will kill it but if you apply and overseed often enough it'll give up the ghost eventually. Lightly apply though because both are non selective. Fire is awesome but such a small margin for error.....

Do you know if you have the common chickweed or the mouse ear chickweed? The common chickweed is considered a winter annual and the mouse ear(Very hairy leaves) is considered a perennial, but I have heard it can act as a winter annual in Oklahoma sometimes. But, being a perennial may explain why it still came up even though you put a pre emergent down. It may have stopped more seeds from germinating though. It seems like I have seen the weed germinate in early spring, die out in the summer, and then germinate again in the fall and then be winter killed or stay green through the winter. About the time you were re-seeding should have been the time to be putting down the Barricade, but we know you cannot do that and re-seed at the same time. Oct-Nov is about the time chickweed germinates. I would try a test patch as philes and morph mention above and you could put down some more Barricade as long as your TTTF is established. If there are any more seeds, it will keep them from germinating. It may also keep crabgrass from germinating early next spring.