how would common problems be resolved?

You will have to be more specific as to what you mean by "common problems".

how would the free market be kept free from being distorted? Surely any structures or mechanisms created in this new society would dsitort the market.

Distortions of the market are those acts that cause a systemic misallocation of resources or an attempted abrogation of economic laws. Occasional or isolated distortions are going to happen in the imperfect world we live in. Let us not pretend that free markets would eliminate all bumps and dips in the economy, or that advocates of free markets claim it would.

how would say the route of a road or the location of a hospital be decided upon?

A hospital would negotiate with a landowner for a location it would like. We basically have roads to everywhere, today. If another road is wanted, its route could be purchased from willing sellers.

How would such a society protect itself?

Protect itself from what?

How would its laws (however few) be enforced? - in essence how would the violence of the state be managed.

There would still be a judiciary and there would still be a policing force. The two are not incompatible with libertarian political philosophy. Because the state is violence, anarchists advocate its complete elimination on the belief that its violence cannot be long restrained. Libertarians seek to put more shackles on it in the hope that its violence can be reduced from its current near totalitarian levels. One step for doing so is stripping the state of its monetary monopoly. Another is by decriminalizing most acts that do not apply to the violation of individual and property rights. Another is disbanding agencies such as the DEA, BATF, IRS, and greatly reducing what police forces can and cannot do.

On what basis would limited resources be distributed?

Be more specific.

How would the weak/defenceless/destitute people be catered for?

What do you mean by, "catered for"?

Im would also be very interested in how people who wish to have democratic involvement in the economy, in how decisions are made etc are to be reconciled with this new society without it being forced upon them.

What do you mean by "democratic involvement in the economy"? What specifically do you think would be forced on us?
"When all has been said that can be said, and all has been done that can be done, there will be poetry";-) -- Issodhos