Can I plant a tri color beech tree 7 feet from house? Can you prune them to keep them on the smaller side?

That is too close to the house IMO.  This is You can keep it lower and fuller through pruning but this will be a constant battle since it wants to be a tree of 40′ +.  I would also worry about such a large root system so close to the foundation. Tri-Color Beeches make a great shade tree out in the open.

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