Moving Products

This is simple to do.  I have had to do it myself.  First , you would set up the new category. Once that is setup, you can go to manage categories and click products.  It will say you have no products assigned to this category.  Click "Add new" products to this category...which will give you the find products searches, select the old category and you will get a list of the products in the category you are trying to get rid of .  You can check all or whatever you need.  If the products are spread across a bunch of categories, use the search feature like SKU or keywords to narrow down the list of products you need to assign to the new category.
Enjoy!

Yes, after you delete the products from the old categories, [which is essentially done the same way: go to manage categories, select the "subcategories", then select "products", and the ones you see should be all the products assigned to that subcategory.  Check all products, then remove checked.]  You can then go back and remove the old categories from the main manage categories area.  If you are using the productcart to create the category links on the home page, you will need to regenerate those as well, also under the Manage Categories.

Yes, see above