Can a Thai national married to a UK citizen enter the UK freely?

If you never lived elsewhere in the EU, EU law has no bearing on the matter and regular UK law applies. Being married to a UK citizen is not enough to enter the country and your wife will need a visitor visa to enter the UK on a visit and a ‘family of settled person’ visa to move there.

If you live in France and your wife has a carte de séjour “membre de la famille d'un citoyen de l'Union” (EU family member’s residence card), she should in principle be able to enter the UK with it if you are traveling together. For a long time, the UK refused to fully implement the relevant EU directive but following a decision by the EU Court of Justice, it relented and issued new guidance that you may be able to use. She might still need a different visa to enter the UK on her own.

Furthermore, a French carte de séjour does not entitle you to live anywhere else in the EU than France but once you have lived in France, the Singh ruling would kick in and you could also use that to settle in the UK.

In practice, I don't think that having used several Schengen visas or owning an house helps by itself (although people have tried to argue things like that in court), it's best to reside in France for a few months, either as a worker or as a non-economically active person. I am not quite sure of the type of documentation you need to show UK authorities to convince them that you did in fact use your treaty rights but paying income tax or holding a French carte de séjour seem like things that would help.

Whether you want to visit the UK under EU rules or use the Singh ruling to establish residency, it would therefore seem useful to first stay in France long enough to get a proper carte de séjour (possibly 6 months or so) instead of going back-and-forth on short-term visas.

Since the whole thing is rather complex and your future life in the UK is potentially at stake, hiring a solicitor familiar with such matters is a good idea if you can afford it.

Caveat: The Home Office has just published their report (dated Summer 2014) called Review of the Balance of Competences between the United Kingdom and the European Union Single Market: Free Movement of Persons which may (or may not) affect policy decisions impacting your question. Not immediately, but possibly in the foreseeable future.

The UK is generally reluctant to honour a French residence permit because the holder is married to you who is a Brit. The UK does not participate in EU residence and/or the Schengen programme. She can apply for a UK visitor visa, and the chances of success are 'iffy' because she's clearly an overstay risk given that you have a home in the UK. Of course, she's entitled to apply, but be aware of the risks.

Two options. The first is what you're already doing: apply for a spouse visa. This lets her come and go for 30 months and it's renewable for another 30 months. After 60 months she can apply for Indefinite Leave (ILR).

The other option is to try the Singh loophole after you have been living with her in France for 6 months. It's a gamble, but if she gets it, it's cost free from there on.

Calling the British Consulate for advice is a waste of time. You will enter a telephone maze of recordings and circular menu options. Consular staff are forbidden (since about 2008) to discuss it with individuals in the first instance. They will route you to their 3rd party contractor who will (astonishingly) charge you by the minute to read their web site to you over the phone.

A BBC article describing the Singh loophole is here: http://www.bbc.co.uk/news/uk-23029195

Adding 27 Feb 2016

Before using the Singh loophole it's wise to consider what may happen if the UK leaves the EU. There is a possibility that those residing in the UK who used Singh may find themselves with a precarious immigration status. The policy has not been drafted, so it is only something to consider. Maybe a tranche of migrants will be required to apply for indefinite leave or maybe they won't. It cannot be predicted and at this point in time it's prudent to consider all the various outcomes. In the event the UK opts out of the EU, they wouldn't get around to even drafting the policy until about 2018 and it would include a transition arrangement anyway.