Do we know the situation in Britain before Sealion (summer 1940)?

The most interesting misconception from that author comes down to dates. The evacuation of Dunkirk happened in May-June 1940, so the map is from before Dunkirk. Before that point, the bulk of the British army was fighting in France. Until the Germans took the French/Belgian coast, there was was nowhere from which an invasion could be launched.

So it is entirely possible that the home defences at that point were quite limited. If your army, and your allies' armies, are fighting in a neighbouring country through which any invasion would have to come, then this is perfectly reasonable.

It would have been more useful to compare this with a similar map from maybe July 1940, when the 330,000 evacuated BEF and allied troops had been redeployed defensively against a potential invasion. The July version would show a completely different distribution of personnel.

To begin with, the answer to the question in your title:

Do we know the situation in Britain before Sealion (summer 1940)?

is, quite simply, yes.

There is a Wikipedia article on British anti-invasion preparations of the Second World War which provides a fairly succinct overview. Detailed reports and maps are held by the UK National Archives (usually in collections with the prefix WO), although I'm not sure how much of this has been published, and a brief Google search suggests that very little - if any - of the material is available online yet.

From July 1940, the defences in the south of England would have been fairly quickly supplemented by Allied troops evacuated during Operation Dynamo, Operation Cycle and Operation Ariel. Further, since conscription had been introduced in September 1939, increasing numbers of new recruits were completing basic training every month and becoming available for service.

The Directorate of Fortifications and Works had been established under the command of Major-General G. B. O. Taylor in May 1940. Most of their records with maps & plans are held by the UK National Archives in the collection WORK 43. Construction of hardened defences actually began in June 1940.

Which brings me to the extract here.

The Wikipedia page on Belton Cooper's book includes the following from Tank and AFV News and the historian Robert Forczyk:

As a memoir, it is meandering and repetitive, far too often wandering away from the authors personal experiences into the realm of speculation. As a history it is lacking, containing no end notes, foot notes or bibliography. And finally, as an indictment of the M4 Sherman tank, the book is filled with so many factual errors and outright falsehoods, it cannot be taken seriously on this count either.

I haven't read the book, but if this passage is representative of the whole, then - harsh as it is - that criticism may not be entirely unjustified.

Firstly, the author claims that following the capture of Liège he found:

"... maps of southern England prepared by the Germans for the invasion of England in June 1940"

But we know that Hitler didn't issue Führer Directive No. 16, which set the preparations for a landing in Britain in motion, until 16 July 1940.

In describing the map of the coast south of Wiltshire, which he says he found when posted to Codford on Salisbury Plain in September 1943, he said that it

"... was dated June 1940"

Which is unhelpful, to say the least. Military maps showing troop dispositions are given a specific date. As described above, the situation on the south coast would have been very different on 30 June 1940 when compared with the situation on 1 June 1940. This may well be a simple lapse of memory, but it does make it very difficult to corroborate his claims.

In fact, the situation that he describes does seem a reasonable match with what what we know about the situation in early June 1940, and General Sir Edmund Ironside's planned defence in depth. This was before the large scale construction of hardened defences really got underway. It seems certain that it must have predated the appointment of General Brooke who took charge of the Home Defence Executive on 19 July 1940. As the Wikipedia article on British anti-invasion preparations of the Second World War linked above observes:

Brooke's appointment coincided with more trained men and better equipment becoming available. Under Brooke, new strategies and tactics were devised. More concentration was placed on defending the coastal crust, while inland a hedgehog defence strategy of defended localities and anti-tank islands was established, each having all-round defence.

(my emphasis)

So, to answer your question from the comments (now in chat), on the basis of the available evidence, yes he may very well have "made up a connection between two entirely unrelated maps that he completely misread".

As a junior officer tasked with the "recovery, repair, and maintenance" of US tanks during the war (which often meant patching the vehicles up, removing any remains of the previous crew, and sending them back into battle), Captain Belton Cooper presumably had other things on his mind at the time.

The final paragraph of the extract:

I have often wondered if the German decision to invade England would have been affected if they'd had a copy of this map when they made their German maps in Liège. A single German division could have completely overwhelmed the pitiful British defenses shown on this map.

is presumably an example of the 'speculation' that the critics described.

As the other answer has already noted, getting that division and the supplies needed to maintain them across the Channel was the fundamental problem.

However, it is also worth remembering in this context that the implementation of Operation Sea Lion was not expected to be possible before September 1940 (at the Berghof conference, held on 31 July 1940, Grand Admiral Erich Raeder stated this could be no earlier than 22–26 September 1940). By that time, any forces that did manage to cross the channel and reach the English coast would certainly have faced far stronger opposition than that described in this extract.

Is the generally accepted evaluation of the proposed invasion as utterly impossible based on the true facts?

The evaluation of the invasion is based on the fact that there was no way for Germany to even land troops. Nobody thinks that the British home defense army could have withstood the German Army on the battlefield. The question could be how much of the German army had to be ferried across to get a decisive victory. But since none of the German army were able to cross the channel, that question is moot.

Sea power was hindering troop (and needed supply) movement across the channel. British air power was hindering the Germans to remove the sea power using their own air force. That is basically what the "Battle of Britain" would be about: gaining air superiority over the channel to destroy or drive away the British sea power to enable an invasion. That failed.

My explanation might be a bit simplistic, but the bottom line is, whether there was a machine gun every 10m or just a guy with a colorful beach towel as far as the eye could see, that is of no importance since no German soldier of an invasion fleet set foot on English soil to be able to put that to the test.