Everything that is designated with the word German somehow concerns the continental Germanic dialect continuum. This designates a region from southern Denmark in the North to South Tyrol (Alto Adige) in the South and from the Belgian coast in the West to somewhere around the Neiße river in the East where ‘descendents of the continental Germic language(s)’ are spoken natively. (The eastern borders were shifted westwards after World War Two.) This dialect continuum is characterised as any dialect continuum is by continuous mutual intellegibility between close variants while there is decreasing intellegibility when comparing variants further apart. Anything inside this continuum can technically be called ‘German’ — but, as always, restrictions apply.
Already in the Middle ages, these dialects could be grouped together into three groups: High German, Middle German and Low German, depending on how strongly they participated in the High German consonant shift.
In 1648, after the 30 years war ended, the Netherlands became independent from what was then Germany (the Holy Roman Empire). From then on, the language evolution started diverging into Dutch on one side and German on the other side of that border. With Belgium also becoming independent after the Holy Roman Empire collapsed in 1806 and being united with the Netherlands after the Congress of Vienna in 1815, the language spoken there was also heavily influenced by the Dutch standard rather than the German one. This explains why the Germanic languages spoken in Belgium and the Netherlands (except for the German spoken in the Eupen-Malmedy region of Belgium) are not considered German but Dutch.
Funnily enough, even though Switzerland also gained independence in 1648, there was less of a desire to separate themselves from German influence. But we’ll get back to that in a second.
The standardisation of German was a long-lasting progress that never reached completion. Most opinions say it started with Martin Luther’s translation of the bible in the 16th century. Luther at the time wrote in a variant that was close to his Middle German dialect. But there was already a shift going on at the time: many cities that belonged to the Low German dialect region had begun adopting what they perceived as ‘the better’ German — a Middle or High German variant. This was a movement initiated by the upper classes to separate themselves from the peasants of the surrounding countryside that would continue to speak their Low German dialects. And since these people had little contact to actual speakers of the Middle or High German dialects, a slightly artificial variant evolved which would mix some northern influences in especially concerning pronunciation. This change was already happening when Luther translated the bible, hence why he may have chosen more southern varieties when in doubt (although he is often quotes with ‘dem Volk aufs Maul schauen’ — listening exactly to what the people say).
Again over time, the standard set down by Luther and others would evolve onwards into what was finally perceived as a common standard German language. But after the Holy Roman Empire ceased to exist in 1806, the different states that belonged to it began taking different paths. Most notably, a successful reunification movement initiated by Prussia culminating in the war with France in 1870/1 made sure that Prussia would be the dominating member of the new Germany by excluding Austria — the only other country of similar size and influence. Switzerland, having been independent from the Holy Roman Empire (and neutral) since 1648, also did not take part in reunification and neither did (whether accidentally or purposely I do not know) Luxemburg. (Including Liechtenstein, sandwiched between Austria and Switzerland would not have made sense and would have caused more problems than benefits, especially due to its minute size.) So henceforth, there were three major countries in which German was spoken: Germany (+Luxemburg), Austria and Switzerland (+Liechtenstein).
This triade meant that the languages in all three countries would evolve somewhat separately. Especially the Swiss variant, which had enjoyed a longer period of separation, became somewhat distinct from the other two variants, both in the fact that the standardised variant is less commonly spoken when compared to dialectal variants and in the different preferred words and somewhat different meanings of words. But even between Austria and Germany there are more differences than some people expect. Therefore, when speaking about German, one needs to recognise that the language is polycentric — much like English — and should be categorised into German German, Swiss German and Austrian German. To add to the confusion, the standard variants are called High German in German.
After the historic overview, let’s focus on understanding and intellegibility. It is true that Dutch nowadays is only mutually intellegible to Germans who live very close to the border, and even they will not understand Dutch people from the far West. Contrary to that, even though they have different standards, a Swiss from the very southern corners of German-speaking Switzerland, an Austrian from the very South-East of Austria and a German from the northernmost point of Germany will understand each other if they use their respective standard variants (Swiss standard German, Austrian standard German and German standard German) — even though these standard variants will still have some local colouring. Thus, the distinction between Dutch and German as two languages makes sense while distinguishing between the other three makes less sense.
Sometimes, even though well within the dialect continuum, the understanding of nearby local dialects can be very difficult even for locals. The Lech river in western Bavaria is one such example: the Bavarian dialects spoken East of it and the Swabian dialects (in the Allgäu variant) spoken West of it are very different and hard to understand for those from the other side.
Thus, after this long explanation, we see that yes, the dialects are typically older than the standard language. But that is no feature inherent of German, other languages’ dialects also predate their respective languages’ standardisation. It also makes sense to subsume those dialects spoken in the German dialect continuum under a common German label, since the difference between them are often smaller than those between German and Dutch or said dialect and Dutch.
Also remember that High German has two meanings. It can be used to refer to the group of High German dialects (high meaning altitude) which took part in the High German consonant shift to a strong extent (only a few dialects took part in it completely). It can also be used to refer to standard German (high meaning elite) in any of the three polycentric variants. To avoid confusion, it is better to use standard German for this meaning.
Taking this into account, yes, Bavarian is a High German (or Upper German) dialect.
Standard German remains a slightly artificial concept. I, a German from the South, will use different words, different grammatical constructions and different sentence structures from somebody from the North even in writing — this is most obvious in salutations and valedictions, though. Compare this to English where anybody from the UK will more or less use the same or very similar words, grammatical constructions and sentence structures in a certain setting.
: Yes, this does include the Netherlands and Belgium. The differences between the German dialects on the East side of the Dutch–German border and the Dutch ones one the West side are small enough to allow understanding and communication.