I was told Hochdeutch and Plattdeutch have a friendly rivalry with one another?
Not really. Low German is pretty much dead due to it being made fun of since the 19th century as a language of stupid people and thus not being taught and discriminated against.
You may find it among old people in small towns.
Because my German friend doesn't recommend Munich or Frankfurt.
I don't see the connection. Neither München nor Frankfurt are areas where Low German was ever spoken.
That said München is a city for tourists and Frankfurt isn't. Unless you are into banking.
Most people I know who've been there highly recommend Berlin.
All foreigners ever recommend Berlin for various reasons.
The city is well equipped to handle tourists as you can get by English quite well compared to literally any other German city.
But this usually results in the tourist not interacting with Germans and having no clue about the country they've been to apart from the Wall and Hitler.
I thought the difference in culture was between West and East Germany- hence the reason I chose Cologne (I was originally going to do Berlin and Munich but was told Munich is boring and overpriced).
The difference in culture is not only East and West. It's not as bad as Switzerland but for example Cologne is in the Rhineland.
Meanwhile on the other side of the state is Westphalia. The people genuinely (friendly) dislike each other and are nothing alike with the people in Rhineland being more open and "fun" and the ones in Westphalia being more like stereotypical North Germans, which they are as Westphalia is traditionally a Low German area while the Rhineland is Central German.
Differences in culture are in states and districts all over the country. Another example being Franconia in Bavaria.
Out of curiosity, will I be allowed to visit the synagogues in Germany?
No.
For that you will need to contact the Synagogue ahead of time and prove who you are, Passport for example.
If they approve you visiting you will then have to show your Passport at the entrance where the police officer will check you.