First, one of the prime exports to the EU is financial services. Major financial flows -- even ones in Euros -- are largely done through London. I can't imagine the remaining countries consenting to allowing that to happen in a non-EU country. Financial services would be excluded from a new agreement. The thing that makes the UK rich is that it was the de facto business and financial capital of the continent, and I don't see how that survives.
Second, the new agreement will need to be approved by every remaining EU country. Some might come on board quickly either because their governments are Euroskeptic (Hungary, Poland) or because they'd rather not upset the current trade balance (Germany) Others have less reason to agree easily. Spain, for instance, wants Gibraltar back and wants money for all of the British retirees its health service is looking after. Ireland will want to be in the driver's seat for any decisions about its land border with the UK. The list goes on. I have a hard time seeing how any of this gets figured out in 2 years.
No it won't, as the EU has become a legislative body governing aspects of people's lives in the UK and it will no longer be the case.
At this point severe trade restrictions with the UK would likely mean even more severe imbalances inside the EU and the eventual exit of Italy, Spain and Portugal.
If you don't want the EU controlling any aspect of the UK, you can't have a trade agreement.
Many of the EU regulations are designed to help consumers e.g. data protection.
So the EU bans toasters and kettles that work: http://www.telegraph.co.uk/news/2016/05/10/eu-to-launch-kett...
Vacuum cleaners that work: http://www.telegraph.co.uk/news/worldnews/europe/eu/11047127...
Some of the other bizarre rules...
It's illegal for bottled water makers to claim it's effective against dehydration: http://www.telegraph.co.uk/news/worldnews/europe/eu/8897662/...
They tried to ban jugs of olive oil at restaurants: http://www.telegraph.co.uk/news/worldnews/europe/eu/10064787...
A salmon smoker had to re-do all package to notify people that it contained fish: http://www.npr.org/sections/parallels/2016/05/03/476064362/b...
There's just a lot of bs. Also the common agriculture policy and common fishing policy haven't been great for the UK.
At last a free market where the curvature of the banana is subject to consumer choice.
I myself prefer the straight banana for optimizing my fruit salad.
The negotiators won't be trying too hard to get a better deal from the EU, because they'll be the ones in power, and if they fail to negotiate a deal they'll be forced to give up that power. Trust politicians to prefer power over principles any time.