Yes. That's what we signed up for: economic and trade relations. Everything that has come after that has never been asked for or voted for here (or in many other EU member states, for that matter).
People who voted in the referendum tell me that it was made clear at the time that it wasn't just a trade agreement, it was a larger project. That was in fact one of the main points of the "no" campaign, particularly emphasised by the Morning Star newspaper and Tony Ben MP.
Why is this a problem? It worked fine for years, and good trade relations are mutually beneficial.
If it is working well why quit?
But the UK isn't asking for a free lunch. It's suggesting that if, say, Germany makes good bread and the UK farms good cows, they trade so everyone can enjoy a tasty burger-in-a-roll for lunch. Plenty of nations outside the EU have this kind of relationship with plenty of nations within the EU today. The UK has, and wants to develop, these kinds of relations with other global trading partners as well.
That isn't correct. Such free trade agreements only work if both countries are on an equal footing, otherwise there will be conditions to keep things fair. For example, Sweden has to abide by most EU rules even though it isn't in the EU, because it wants a free trade agreement. If it didn't abide by those rules it would be free to, for example, treat employees significantly worse and thus give companies an economic advantage, or have the government support failing industries which EU governments cannot. Thus there would be conditions to enable EU businesses to compete fairly and not be undercut.
So our choice would be to either accept most of the EU rules without having any say in them, and thus not be able to get rid of one of the most hated (and also beneficial) aspects of membership, or to forget about free trade and pay duties on products and services exported there. UKIP kind of acknowledges this and claims that the benefit of being able to trade with fewer restrictions with other countries would make up for it.
Note also that when people say the UK wants to develop relationships with other global trading partners, what they mean is that they want to reduce conditions and wages for employees to the same levels as those economies. Why should they pay you more than some guy doing the exact same job in China? How can they compete with that?
The UK is a net importer with most of its major trading partners within the EU. Financially speaking, there is probably more benefit to those nations if they preserve good trade relations with the UK than the other way around, but both sides benefit greatly.
Doubtful. There is the matter of principal, but more practically other countries will see it as an opportunity to improve their own positions. That's what the UK wants to do, it should expect the EU to do the same. Clearly there won't be a simple continuation of the existing arrangement because as you point out, the UK wants to ditch many of the rules on which that arrangement is reliant.