How should I approach my landlord about doing basic repairs on their house?
I have been a renter in a similar situation to you.  The types of projects I did were major landscaping, installing hardwood floors and trim, replacing old appliances, plumbing repairs, and pest control.

In general you are going to have the easiest time with low cost, low risk work.  By low risk I mean risk of messing up something serious that ends up costing the landlord a lot to fix, or injuring yourself or others (immediately or for future tenants).  

Start with a smaller projects, like the pruning and filling cracks.  Describe the work you'd like to do and get approval in writing.  Include in your agreement a way to be reimbursed for minor expenses by turning in receipts when the work is completed.  If the landlord is not local, be sure and send pictures of completed work so you can show how nice it is.  This will build their confidence for allowing you to do future work.

I never had any problem getting reimbursed for out-of-pocket expenses that were minor or agreed to in advance.  I think you may have more difficulty getting paid for your labor.  I never got paid for mine outright, but I did negotiate a lower rent.