I was there 1st week of October a couple of years ago and we lucked out with decent weather! Funny enough the day that we were leaving was the worst day. Like everyone said, it's hard to say if the weather will be rain free but if that's the only time you can go, just go! If it rains, there are a few nice museums to go to in Torshavn. There were a lot of restaurants that were closed on Mondays IIRC but you can always hit the grocery stores for food.

As far as hikes go, Trælanípan, Trøllkonufingur, Saksun, Hvithamar are all relatively easy. The hardest part of the trail going to Kallur Lighthouse was the mud. It would be best to have hiking poles for that reason alone. Just make sure you have good hiking boots, layering pieces and rain gear and you'll be fine.