From what I have been reading of your question, you do have a bit of a drainage problem on your site.
An immediate fix would be a 'dry well' off to the side of your lawn.
Dig a big hole; 2 to 3 feet deep and 6 to 8 feet in diameter. LIne it with landscape fabric, dump in river rock, drain rock, cobble they are all the same thing to fill the hole, cover with landscape fabric, then cover with whatever surface material you've been using. This gives excess water a place to go and be absorbed into the ground, more slowly. Without causing flooding. Very inexpensive solution and extremely sensible!
Pavers or 4" of crushed gravel would work very well along the side of your home. Don't think about 'building areas up', think about scraping soil off the surface to promote drainage of every drop of water that lands on the soil around your home: Every bit of land around your home has to slope away from your foundation. Period.
Usually, you are not allowed to drain the water coming off your property onto your neighbor's. I would also look up the laws, rules of your city or county buildings and code as well as your 'development'.
I recommend you talk to your neighbors about their basements and past dealings with drainage issues, talk to Buildings and Code about the rules concerning drainage. Most places will not allow a home owner to drain excess water into the storm drains.
For your wooden structures, you need a minimum of 2" below the wood of the fence. 4 inches is better. There should be at least 4" between the bottom of the siding of your home and the soil.
Finally, I would check out the foundation, your basement and crawl space. Ensure your foundation has a perimeter drain, back filled with drain rock, drainage pipes covered with landscape fabric. There should be asphalt emulsion on the concrete of your home's foundation between any soil and the concrete of your foundation.
Drainage makes or breaks the value of your home and landscape! Not something one dismisses to do for later.