This question is way too broad, but address #1, i.e. why the EU regards those as inseparable if a single market is to be achieved, the answer is that workers are regarded as productions units so they need to be interchangeable / exchangeable across the market. (There's a quote from the EU founders on this, bear with me while I find it. I've answered this q at this level of precision, because it has several pending close votes [probably due to its polemic tone].)
Now the rest of your question(s) mostly confuse free-trade agreements with a single market, e.g. neither the EU nor Canada claim that they've achieved a EU-Canada single market. Keep in mind that although EU and Canada do have a FTA, Canadian trucks (or ships) don't simply roll in the EU with their merch without any border checks. Even on the level of trade in goods, there's no customs union between EU and Canada either. As a third party, you can't export something to the EU (or Canada) and have your merch then seamlessly move across the EU-Canada border in the aftermath (as it happens inside the EU single market).
And given your comment below Jontia's answer, you seem to assume that "freedom of goods" is the same as an FTA, but that's not how the EU sees it, nor did the ever see it just like that, but (at the very least) in the tighter sense of a customs union:
Originally, the free movement of goods was seen as part of a customs union between the Member States, involving the abolition of customs duties, quantitative restrictions on trade and equivalent measures, and the establishment of a common external tariff for the Union. Later on, the emphasis was placed on eliminating all remaining obstacles to the free movement of goods, with a view to creating the internal market.
The EU and UK won't have a customs union anymore, so the exception you posit at your #3 doesn't exist in that sense. A better question in this regard would honestly be Turkey, which does have customs union with EU on some level.
And as an EU document explains this in relation to the EU-UK negotiations:
There are also major differences between FTAs and the customs union and single market that an FTA could never
match. Most notably, even though most, if not all, tariffs are reduced to zero, goods are still subject
to border checks for customs verification. Among other things, exporters must demonstrate the
origin of manufactured products that justifies the application of preferential tariffs. The exporter
must also demonstrate that goods comply with EU standards, and goods are subject to verification
by importing countries at the border.
If this sounds like small potatoes, it's not. It took the EU decades to achieve this level of
goods-market integration just in terms of eliminating intra-EU border checks, despite this being a goal in the 1957 treaty. You can read (for instance) a 1985 EU document on the efforts to complete this by 1992, and ponder on the complexities that were involved... or what the EU-UK border might revert to. In that 1985 document, the EU Commission was estimating the that the frictions then caused by intra-EU checks were equivalent to a 5% tariff on all EU manufacturers.
As for the EEA/Norway approach, a 1998 EU parliament report says that
the EEA does not constitute a fully "frontier-free" market, nor a true customs union. Yet it provides for fundamentally improved free trade. [This despite the fact that] Over 80% of the Single Market legislation (some 1,500 directives, regulations, decisions at the time of signature of the Agreement) applies within the EEA.
I'm sure some would describe that as (virtually) being in the Single Market, but I though it's interesting to point out that some EU reports disagree with that unequivocal formulation.
We may be getting a bit far afield here, but on the level of theory of economic integration, the following categorization has been proposed:
three sets of policies are involved in the
process of economic integration:
(i) the elimination of border measures applying to imports into one member country from another member country;
(ii) full National Treatment of beyond-the border measures applying to imports into one member country from another member country; and
(iii) harmonization of measures across member countries.
As discussed in that paper, more recent FTAs (e.g. NAFTA) actually cover (ii) fairly substantially, in fact even the WTO rules do it to a good extent. But FTAs generally do not extend National Treatment to labor markets, so in this regard they fail (ii).
As [fairly trivial] example (given in the paper), a national standard that requires things to be measured in inch satisfies (ii) but not (iii), and so it's still a [non-tariff] trade barrier to manufacturers from countries that use a metric system. The Treaty of Rome didn't actually cover (iii) too well, this actually being a major reason for the (1992) Single European Act.
As far as economic theory goes, a single market is fairly easy to define:
To give precision to the concept of a single
regional market, economists have defined a single
market as one in which the Law of One Price must
hold in all goods, services and factor markets; see,
for example, Lloyd (1991) and Flam (1992). That
is, there should be a single price in the regionwide
market for every tradable commodity and factor,
expressing all prices in a common currency and
adjusting for the real costs of moving goods or
factors between locations. This definition allows
for the real costs of moving goods or factors from
one location to another.
This definition of a single market can be applied
to any set of countries. This may be a region
comprising several countries as above or just one
establishment of a single market, therefore,
is much more demanding than the establishment
of a common market.
However, when defined in this sense (no spatial arbitrage),
the notion of a single market is much more strict than even
what the EU has (actually) achieved. Such a definition even makes some federal countries fail the test, because e.g. of differential regional taxation:
Some countries with a federal structure would not be regarded as a single
market in this sense because state or provincial laws, regulations, taxes and charges create price differences among the states or provinces.
Nevertheless, one can see why various transnational measures may be needed to progress towards that ideal e.g.
A single market crossing national borders
requires the removal of all border restrictions and
full National Treatment with respect to taxes and
other state charges and regulations. These steps
may need to be supplemented by the har
monization across national borders of laws and
regulations which otherwise prevent a single price
from ruling among the countries. In goods
markets, these standards include industrial
products, health and safety of persons, and the
environment, policies relating to particular sectors
such as industry or transport. The laws include
business laws that differentiate between foreign
and domestic supplies. In labour markets, full
National Treatment requires measures such as
the recognition of foreign labour market
qualifications. In capital markets, it requires full
national treatment with respect to taxes and
business laws and regulations. It implies the
absence of such measures as performance
requirements that apply to foreign-owned
enterprises but not like domestic enterprises.
The paper then discusses how even allowing for different currencies may actually
break a single market (in the strictest sense) because the behavior of
risk-averse agents implies incomplete pass-through of foreign prices to domestic
markets when multiple currencies are in use.
But the paper grants that despite failing the theoretical ideal
the EU is clearly the RTA that has progressed the furthest towards complete integration