Under UK VAT law which of the following items are zero-rated and why: spaghetti bolognese, gazpacho soup, a toasted cheese sandwich, magic mushrooms?

Ok, apparently food items in the UK are exempt except the following: alcohol, meals out, hot takeaways and certain foods such as ice cream & confectionery. However, there was an instance where by toasted sandwiches were considered exempt because people argued that the sandwiches were not toasted so as to be served warm but "to release the flavour of the ingredients and to make the bread crisp".

So, the spaghetti being hot and most probably a meal out would have the tax on it. The soup, being a cold one wouldn't have the tax presumably, but it could well be a meal out item...

Now, the magic mushies-- As they create a hallucinogenic "trip", they are subject to VAT at 17.5%, because they are considered a drug not prescibed by a doctor (usually). But there are arguments because at the same time, the Home Office has said that the sale of the mushrooms may be illegal which, if true, will mean they are not subject to VAT at all.

A good helping of yay for that Peasy.
In Britain, "prepared" food is taxable, but zero-rated if the food is cold when it reaches the customer. Toasted sandwiches are zero-rated because the purpose of toasting relates to flavour and texture and not to the necessity of the food being hot when eaten. Also, as you say, according to HM Revenues and Customs, food that is hallucinogenic is not exempt.
So the gazpacho and sandwich are zero rated, the other two are standard rated (17.5%).

I'm not British, so...what does zero-rated mean?

There's both No VAT and Zero Rated VAT. The difference is that, while the customer doesn't pay any VAT on either, if an item is zero rated the supplier can claim back the VAT he has paid when getting or making it. If there's no VAT, he can't. VAT works as a chain - it's a tax on the value added to a product. So the person that digs it out of the ground pays VAT on the stuff used to produce it, and charges VAT on the product when sold. Then claims back the difference. The next person to handle the stuff has paid his VAT, and can claim that back against the VAT he charges when selling it again. And so on, down to the final user - thee and me. And we can't claim anything back. Sounds daft, but don't forget this is taxes we're dealing with. And legality of a product or service doesn't remove the obligation to pay the required taxes. They didn't get Al Capone for murdering anyone - it was the tax people that got him. The problem for people in illegal activities is that if they declare their income from them, they could find themselves in worse trouble than for evading taxes.

These was recently a big question mark above the VAT ratings on shoes. Children who have abnormally large feet (for example, being 8 years old and brandishing adult size 5 feet) are having to buy shoes designed for teenagers, but these, of course, are not VAT exempt.

The arguement really was that the shoes were being bought for children to wear, and secondly, when does a child become a teenager? This means that teenagers with unusually small feet were VAT exempt but you children with large feet were not.

And believe me. When buying trainers, that 17.5% can make a hell of a lot of difference to the price you pay.