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To some extent, the answer depends on how much you'll be removing. Minor trimming and removal of dead branches can be done anytime. Some say not to prune maples in winter or early spring since it will cause sap to bleed and the loss of the nutrients in the sap. Pruning right after the maple leafs out seems to be the best time. Summer is OK as well. Fall and early spring must be avoided. I suggest you go to your local extension service website for information on your locale. (search for [your state] extension service) In some places fall is right around the corner and it may be too late for summer pruning. Here is what the Arbor Day Foundation has to say: https://www.arborday.org/trees/tips/when-to-prune.cfm

Fall is a great time to plant red maples. I think you should go for it this year. Trees planted when they're smaller often do better than those planted when they're larger.

Don't fertilize the tree during the first winter to avoid stressing it; wait till next year. Here are some articles with advice on planting maple saplings/seedlings and other trees:

https://www.gardeningknowhow.com/ornamental/trees/maple/care-of-red-maple-trees.htm
http://csfs.colostate.edu/seedling-planting-guide/
extension.psu.edu/publications/ee0127

That is one of the many varieties of the red maple! Those that you see are seeds and will soon fall to the ground. This will give you more information on tree: https://www.gardeningknowhow.com/ornamental/trees/maple/care-of-red-maple-trees.htm