It's not about written or spoken German. It's about different text types.
Maybe you know the concept of text types. The language people use to write scientific articles is different from the language you find in love romances. (In scientific articles there are almost no personal pronouns. Love romances use much more adjectives.) Also the manual for a vacuum cleaner is written in a very different style than a report of a sports event.
But not only the style is different. Some text types even can have their own grammar rules: The verb, that in German statements is always at position 2 of a sentence, moves to position 1 in jokes (»Geht ein Mann zum Arzt.«) And cooking recipes use a feature with the name »adhortativer Konjunktiv« which is hard to find in other contexts. (»Man nehme eine Priese Salz.«) Even when it is used in an non-cooking context (3 book titles: »Man nehme einen neuen Mann«, »Man nehme einen Geigerzähler«, »Man nehme einen Sonnenstrahl«) it always strongly refers to recipes.
But although the name of this feature is »text type« it applies not only to written language but also to spoken language. The way how native speakers build sentences is different when they sit at home on the couch and talk to their partners or when they stand at the lectern and deliver a keynote speech. The grammar of the sentences a person says who wants to sell you a life insurance is different from the grammar from someone asking for the way to the next train station.
Some text types are used in written language only, some others can be found only in spoken language. (A written text can always be read aloud and a spoken dialogue can of course be written down, but this possibility of transformation into the other system does not contradict the original division.) But the difference between written and spoken language is somethign that can't be overseen: One class is produced with hands and received with eyes while the other is produced with mouthes and received with ears. Because this difference is so obvious and clear, we tend to blame it for differences in grammar. In reality, however, different types of texts have different grammatical rules.
Q 1a: How different are Spoken and Written German?
As shown in the previous section, this question starts from a false premise. The difference is not in how language is produced or received, but in what it is about.
Q 1b: Should they be learnt bearing these differences in mind?
Yes. If you want to write love romances, read love romances and write your novel in a similar style. If you want to write cooking recipes, you better should write them similar to other existing cooking recipes. If you live and work in a German speaking country and you want to talk to your neighbors and collegues you should listen how they speak with each other and try to do it the same way.
Q 2: Does using Präteritum and phrases like "Er hat mir gesagt, er sei nicht geschwommen" by speaking sound strange?
This is not »Präteritum« (preterite). This is »Perfekt«:
Perfekt: Er hat mir gesagt, ...
Präteritum: Er sagte mir, ...
There are also some possibilities for the relative clause in colloquial speech:
..., er sei nicht geschwommen
..., er wäre nicht geschwommen
..., er ist nicht geschwommen
..., dass er nicht geschwommen sei
..., dass er nicht geschwommen wäre
..., dass er nicht geschwommen ist
You can find both preterite and perfect in the main clause in spoken German and all 6 variations of the relative clause. Some of them are considered to be wrong in written German, but this again is more a question of text types. If you are in a dark bar in a cheep district, surrounded by drunken hooligans, the version that is correct in written German can be terribly wrong when you say it there.
The Influence of Geography
The point is: It depends strongly on the geographic region where you say this sentence. Geographic differences is a topic on top of text types.
There are some invisible border lines across the German speaking area: The Benrath line, the Speyer line and some other different border lines. These lines are isoglosses. An isogloss is a border between variations of a language or between language features. And among many other features, these lines separate regions where people prefer preterite over perfect from regions where people prefer the other past tense.
The preterite region is in the south (mainly Austria, Switzerland, Bavaria, Baden-Württemberg) while the perfect region is north of these lines. So when you say that you didn't know something, you can say »Ich habe das nicht gewusst« in southern regions but »Ich wusste das nicht« in northern regions.
The regional differences are on top of text types, so you would expect that you find them also in written language, but there is another circumstance:
The Influence of Media
My grandfather lived in the south east of Austria and wrote books in Austrian Standard German and even in the local dialect. The Austrian crime fiction author Wolf Haas also writes his books in Austrian Standard German. When you read these books, you will notice that, when ever something from the past is reported, they use perfect, because this is the preferred past tense in southern regions (»ich bin gegangen, er hat geschlafen, ...«). But when you enter a book store, even in Austria, take any arbitrary book, open it and start reading, you will find preterite (»ich ging, er schlief, ...«) in 99.9% of all books you try.
Here is why:
Germany has 83 million residents (more than 70% of them living north of the preterite/perfect isogloss, in the preterite region), Austria has 8.9 million and Switzerland has 8.6, but only about 5 million of them speak German. Books printed in Germany are written in German Standard German and read in Germany, Austria and Switzerland. They use the preferred tense from northern regions, which is preterite (»Ich ging, er schlief, ...«).
If someone writes a book in Austrian Standard German or Swiss Standard German, it will not be accepted in the two other countries. So the best strategy to write books that you want to sell in the 85%-market Germany is to use German standard German. And this is why (with some really rare exceptions) authors form Austria and Switzerland also use preterite in their books. And so not only in (almost) all books preterite is the favored past tense, but also in magazines and newspapers. Even in those that are not supposed to be exported. And this is why you find preterite in almost all written texts.
Q 3: If the answer to the first question is "yes", then: which other characteristics are typically "written German".
As said before: There are characteristics that are typical for different text types and characteristics that are typical for geographic regions. I would not really nail it down to written vs. spoken.
I've talked about differences in text types a lot in the sections above. Geographic differences also find their way in written texts. Nobody who was socialized in Austria would use the noun »Joghurt« with a masculine article (»der Joghurt«) as it is usual in German. It's a neuter noun in Austria (and maybe also in Bavaria, i'm not sure about that): »das Joghurt«. And you will find das Joghurt also in Austrian newspapers and magazines that are produced for the Austrian market only. But when an Austrian author wants to talk about yogurt in a book that shall be sold in Germany too, the Author must write »der Joghurt«, even if the publisher is in Austria. This might let you think that the different gender is a matter of written and spoken German, but in fact it is just a matter of geography.