If I buy a product and attach it to my invention, is that infringement?

The answer depends on what sort of product you are buying and whether or not it has any restrictions on its use, further adaptation and the like and whether or not you will be reselling the combined product. Such restrictions are common with various license agreements that come with the sale of products, particularly software with regard to patent rights and copyrights. Additionally, there could be some contexts where trademark rights may be violated if you modify the purchased product and resell it under a new trademark that might cause confusion as to the source of the goods. If on the other hand you are simply purchasing a product that you want to attach to your new concept/idea for you own personal use, it is unlikely that an infringement issue will arise. As always, you should consult an intellectual property attorney so that he or she can fully explore your facts and intended use in order to properly advise you of your rights and possible liabilities.

Not at all.

Patent Exhaustion (the first sale doctrine) generally means that if you buy an authorized product that is covered by a patent, you can then use it and/or sell that particular physical item without violating the patent. If you do a search for patent exhaustion, you can find out more details.

Yes, if the item purchased was from an authorized seller (e.g., not a knock-off copy) - you can use the device in your invention. The first sale doctrine, is a common law patent doctrine that limits the extent to which patent holders can control an individual article of a patented product after an authorized sale. Under the doctrine, once an unrestricted, authorized sale of a patented article occurs, the patent holder's exclusive rights to control the use and sale of that article are exhausted, and the purchaser is free to use or resell that article without further restraint from patent law.