Hard to know for sure but countries that already have a trade deal with the EU are bound by EU laws and have signed up to the EU freedom of movement agreement
Disclaimer: I don't believe we'd be likely to get a free trade deal with the EU without allowing freedom of movement. Though it might just be possible after a decade or two of negotiation?
But, Canada have a trade deal without. Most of the EU's trade deals don't involve freedom of movement. It's a political decision to insist on it around here, which I'd say is unlikely to change. Though there's talk of a free trade deal with North Africa, and I bet there won't be freedom of movement on offer there But then Norway and Switzerland are both in Schengen and the Single Market (well only partially for Switzerland). Which is different to a free trade deal.
Although Norway and Switzerland are agreed to free movement (or disagreed in the Swiss case), they don't have to harmonise access to benefits in the same way. So were we to leave we could refuse to pay in-work, unemployment and child benefits to people we don't choose to, and I assume the same might be true for the NHS, we could make people pay an insurance for that too, until granted "permanent residence status" of some kind. There are many options, which we might use to reduce low-skilled migration. Although one of the best ways to cut the migration numbers to the UK would be if the fucking Eurogroup would sort their shit out, and actually fix the Eurozone! But there's not much sign of that happening any time soon. The latest IMF prediction in the bail-out debt sustainability analysis, is that Greek unemployment will fall from it's current 1930s Depression level of 25% to about 12% by 2040! Now that's how to run an economy! And quite a lot of that fall is expected to be achieved by emigration. Well with youth unemployment at 50%, wouldn't you?
Also, Switzerland and Norway are only signed up to something like 20% of EU rules by being in the Single Market. That still leaves them the ability to sign their own trade deals, and out of the Common Fisheries and Agriculture Policies. As well as safely out of the Euro, and the Foreign Policy and tax harmonisation stuff.
Any country that exports into the Single Market is subject to large chunks of the same regulations, in order to make their products compliant. In the same way that you have to comply with lots of US laws in order to trade in the US.
The upside of being in the EU is that you can influence the rules. Subject to getting outvoted. The downside is that you're subject to a lot more of those rules than any flavour of leaving (either EEA or even fully out).
In my opinion the only sane option on Brexit is the Norway option. Nothing else can be negotiated in a sensible time. But done as a temporary deal with more negotiations to slowly follow. That means we pay in a bit less cash and get a bit less inward migration, in the short term and that gives everyone time to calm down and come up with a sensible agreement.
Either that or the other governments accept a 2 speed Europe, which is what we actually have anyway with some countries unwilling to join the Euro. If they'd done that, then Cameron would have won the referendum easily, but they only gave the minimum concessions that looked likely to work - and that may now backfire spectacularly. To be honest I still think it's more likely than not that the Euro will implode when the next recession comes (not that Souther Europe have ever really got out of recession) - and quite possibly take the EU with it. But the Italian economy is now smaller than when it joined the Euro, with deflation, youth unemployment of 40%, extremely low growth and a huge government debt which it can only service with some growth or inflation. Plus a half-collapsed banking system that the new Eurozone banking rules make it impossible to restructure. That's the quiet crisis currently going on. Spain, Cyprus and Ireland are doing OK-ish, Portugal is too small to matter if/when it goes wrong again, and Greece is a shameful stain on the reputations of the governments that have fucked it up for short-term political gain.