Lasagna gardening is meant to be used when preparing a new bed. You apparently are referring to applying a mulch that the roots will grow up into. For rhododendrons that is not a good idea. The purpose of the mulch is to conserve moisture and keep the roots cool in the summer and insulated in the winter. If the roots grow up into the mulch, then it does neither. If the mulch is too deep it will kill the plant. Another worry, peat moss is not a mulch, it is a soil amendment. So I don't think that it is wise to try to think of lasagna gardening around an established rhododendron. You only have to worry about the rhododendron roots out about to the drip line of the plant. So if you want to remove the sod or whatever, all options are open. It is just the area inside the drip line of the rhododendrons where you should not disturb the shallow roots. I foresee another problem. If you build up the area outside the rhododendron drip line, the area around the rhododendron becomes a depression and water will puddle there. That is very bad. Rhododendrons should be mounded up from the surrounding soil. So I would say that removing the sod outside the rhododendron's drip line is much better than doing a lasagna or something that builds up the surrounding soil. If you apply over 2" of mulch, the mulch starts to be a problem. It is recommend to only use about 1" to 2" of mulch. Mulch is there to hold in the moisture and keep the roots cool in the summer and insulated in the winter. If you use more, it can smother the roots or as mentioned previously stimulate the roots to grow into the mulch. A heavy deep mulch prevents the aeration of surface soil / roots and death quickly follows. I think a large piece of plastic could be disastrous. If the sun hits it, the plant is literally toast. When you are done, it is big and awkward and contaminated. When I spray near a rhododendron, I carry a piece of cardboard. I place the cardboard between where I am spraying and the rhododendron. I must mark which side of the cardboard might have roundup on it, or I could be making the problem worse. If roundup does touch a rhododendron leaf, either wash off the leaf with clean water or remove the leaf. When using roundup, never use it on a windy day. Use a course mist that won't be affected much by any wind currents. Use a low pressure in the sprayer. A higher pressure tends to atomize the spray more. It is easier to get a course spray with a low pressure. Don't use the new "Roundup Extended" around a rhododendron. It has a second chemical that is designed to provide residual weed killing for use around sidewalks and driveways, but not to be used in gardens.

Botann's point is well taken it does supply good fast shade but in my experience they NEVER get taken down once they get bigger until the storms lay it on your house large branches in your yard and an ugly possibly mishapen tree.Folks always plan to but by the time they get around to it the tree is 40-50 feet tall and not easily removed.The Silver maple in our area can grow 10-15 feet pr. year up and out( mine do) without much problem a tree your size will be probabably 20 ft tall maybe taller and probably 15-20 ft. wide in two more years if your soil is half as good as mine... Remember the larger the tree the more it will cost to have taken out especially one that close to your abode. Of course this is your decision David

"Foundation plantings" are a landscape design convention held over from Victorian times when shielding or disguising bare stone or concrete foundations was popular/necessary and popular landscape architects of that time promoted the idea of 'grounding' the house to the landscape for a more natural look. Like the omnipresent lawn, they may or may not have relevance to today's landscapes. And it is a matter almost entirely of personal taste. Certainly lot sizes have shrunk significantly over the years so that the beds that defined a foundation planting are now so tight and restricted that almost nothing other than annuals and a few perennials fit. That appears to be the case in at least part of the OP's garden, where the concrete walkway only allows a very narrow space for any sort of planting bed. But if the foundation bed is designed to be generously deep enough (a 6' minimum - deeper is better) then it allows for planting some evergreen material (mixed with deciduous if desired) to compliment the house and welcome visitors, provided that it is of proper scale. And if well-designed and with proper consideration to plant selection, foundation plantings pose no issue to either the structure or the foundation. btw, the design lingo now popular for this front of the house planting is 'the entry garden' :-)

It probably won't hurt the tree as far as having a few less branches is concerned. But depending on your location and tree orientation, the higher you go with the branches, the more sun exposure the sun will have on the ground in that area. There will be more heat and dryness. I always like to raise a crown over a period of time, maybe two seasons. This allows both the top leafy parts and the bottom root system to adjust to the abrupt change they experience by doing this all at once. Be sure to use a layer of good wood mulch around the tree to help protect the root system from the sun.