If 1&1uk Offers plans which include MySQL databases, and you are planning to run an application which requires at least one MySQL Database, then you should have gotten the minimum plan that supports your needs.
Before you consider whether you need a different solution, consider whether you have what it takes to run a business. If 1 pound 50 pence per month can keep you from having the bare essentials - you may not be able to afford this business.
Consider also your other recurring expenses. You'll need your own SSL certificate. This need not be expensive to begin with - but when even using Paypal or another hosted payment page you will need to secure account creation and history pages, as well as Admin pages. This can increase your sales conversions by 50% or more. Not sure how pricing works out in the UK - but in our US based market we sell these for as little as $14.
This might require you to pay a bit extra for your own dedicated IP Address. This is not like a dedicated server, but if you have one, you won't show up as just another listing on a shared server. This reduces your exposure to hacking and the risk of your mail being confused with the porn sites on the server (if any) and blocked by blacklisting services.
Password managers can be had for free - KeePass is a good one. This will help you change passwords on a regular basis. Do this for at least your FTP and control panel accounts - and the billing system if it is different.
These are the basics of web commerce. Don't even think of starting without them.
Alternatives to osCommerce? There are quite a few. Magento, Zen Cart, CRE Loaded, Prestashop are good places to start looking. They all require a good database server, just like osCommerce.
If your dad already has experience with osCommerce - use it! I routinely load data on thousands of products into osCommerce or osCommerce based stores and deploy at least one page per product in under a second each.
If you can do that with Dreamweaver, contact me. I've got a job for you.