1000' is incorrect, certainly not for 100BaseT and absolutely not for 1000BaseT (gigE). 10BaseT, perhaps. You will not get reliable Ethernet over that length. There's fundamental issues of RF latency that will not work at that length. You might get the link lights to engage, but if you go back and track the error rates on the packets they'll likely be unacceptably high.
A router in bridge mode, for what? Yes, for wireless they're handy, dd-wrt can definitely do the job. There's two ways to approach it, run a WiFi network at the house and then use a router at the outbuilding to act as a client. Or use two routers, one on each end, so that all they do is provide a point-to-point network. The latter approach requires a second unit, of course, but may be more reliable. That and your traffic between the buildings won't be on the same WiFi channel as stuff in the house. If you want additional WiFi coverage in the outbuilding you could use a two-radio router, one to handle the point-to-point and the other to provide local WiFi. The configuration for this sort of thing gets a bit complicated and not all router radios will do as effective a job as others. It's very hard to pin this down as there aren't a lot of people trying to use them this way. Me, if I'm going to re-purpose lower-end routers like this I prefer to keep each of them configured for their purpose and just use more of them, instead of trying to force fewer things to do more. I know enough to be able to do it, and I prefer to avoid it.
Having installed fiber myself, it's not really much more delicate that CAT6 during installation. Which is to say CAT6 likewise requires being mindful not to break the conductors inside the jacket. There are specifics with how connectors are put on the end, but the kits for gluing them are simple. You do need to be prudent about how you handle the fiber strands at the ends. That and make sure they're effectively supported/secured into their converters.
