Yes, your customers can send payments to your paypal account without a shopping cart. You don't need a shopping cart to accept Paypal payments.
You will need to create Buy Now Buttons from your paypal account. You may would like to learn more information about Buy Now Buttons, here is the link: https://www.paypal.com/pdn-item
Buy-Now-Button will take your customers to Paypal's secure shoopping page. In that page, your customers will be able to make their payments. They will be redirected to your website after completing their payments. You will also have full ability to select the exact redirection url(Paypal will redirect your customers to that url after succesful payments).
There are very nice open source PHP shopping softwares. You can use them. That would be best option. If you don't know anything about websites, you may would like to hire someone to do everything for you.
There are some websites which allows you to run your online shopping website. In example: http://www.shopify.com/
I have read your story. You would like to track your stock. You don't want to accept payments for the products which you don't have on your stock.
You can track your stock with Oscommerce. Most of shopping softwares are able to track product stocks. You will not need to remove products from sale, because your software will already remove the products if they are out of the stock.
I hope you would find the best option for you.