How long does my landlord have to fix the fence and pick up a fallen tree?

If the landlord does not repair the fence, your best best is to sue him or her for a court order requiring the repair and also potentially seeking some compensation for the time you have not had the fence. That is because you can only break a lease due to a failure to repair or maintain the property if the failure significantly impacts the property's habitability--that is, the failure  makes the property essentially uninhabitable. A broken fence and an inability to leave your dogs in the yard would not even begin to approach that level of impairment.

As for fixing it yourself and deducting the cost from the rent--the "repair and deduct" remedy is also typically available only when the repair must be made to maintain habitability.  Tenants generally cannot withhold or deduct a portion of rent for repairs which do not impact habitability. So again, despite the inconvenience, this situation is most likely not serious enough to support doing this.