1. How do I add a product to a cart from a non-cart page
Use OrderAnywhere links that you can get from Merchandising > Order Anywhere
How do I clear a cart from a non-cart page
You don't. The cart can only be cleared from the shopping cart screen using the Empty Cart option.
How do I modify the available options (like remove shipping options) based on product
You cannot modify shipping options based on a particular product. You can say whether a product has shipping charged on it but you can't specify different options based on what the product is. In ShopSite Pro there is the ability to have a custom shipping module that returns shipping costs to the cart. You might be able to do something with that to exclude some shipping options if a particular product is added to the cart but it would affect the shipping options for the entire order (all products in the cart) and not a specific one. Using the shipping api you have to provide all shipping options and do the calculations for them in the api. Sample code can be found in the online help, but unless you are an experienced programmer, or are willing to hire someone to do it that is, it might be a little complex for most merchants.