How much % do each of the Jewish denominations make up the Israeli Jewish population?
Israel today has a mainly secular population although that isn't how the future is looking. The numbers differ but around 12-15% of Israel is considered Haredi and about the same number would consider themselves MO/Conservative. But the arrow is pointing towards a Haredi future. Through things such as a higher Haredi birthrate and the baal teshuva movement Israel is becoming a more religious country. This election the Haredi parties managed to win 16 seats which is a huge uptick from the last election. It's simply undeniable that we are gaining more power. Whether its 50 years or 100 years everyone knows soon enough Israel will be a majority haredi country.
Is there any religious difference among the dynasties besides location/leadership?
Yes there are HUGE differences between the different dynasties in so many ways. Just look at the topic of kiruv, dynasties such as Chabad and Breslov choose to put real effort into kiruv while dynasties like Satmar vow to stay insulated. Also look at the approach to Zionism between the dynasties. Dynasties like Toldos Aharon and Satmar don't even recognize the State of Israel while a dynasty like Chabad-Lubavitch is pro-zionist. Their attitudes to religion are also very different. For example the Breslov are all about serving G-d with joy while the Satmar are extremely serious.
Before European Jews started being more secular and reduced religious observance in the past 100-200 years, were all Jews in Europe as religious and secluded as modern-day Haredi/ultra-Orthodox (apologies if you consider those 2 separate things) or was it always a similar spectrum like today?
To explain the history of Jews couldn't be done in one comment. Haredi Judaism is like an unbroken chain of tradition from Sinai. Also there is no such thing as "ultra-orthodox". It's just an insulting term used by the secular jews to put down the orthodox.
Also, I read on this sub of Israelis/Jews worrying about the Haredis population growing too much and taking over the country and influencing the laws. What do you think about that?
This sub is representative of the true Israeli population which is mostly secular. So what the secular thinks are good are not good for the Haredi. We don't want to serve in the IDF with a gun or to go to college. What we want is to serve G-d. It won't be very long until haredi jews are 20-30% of Israel, maybe just a few decades. That will be a great time in Israel. We will finally see the end of the IDF forcing yeshiva boys to serve and all public transport will be closed on shabbos. The values of Judaism will be the law of the land when the Haredi have control. It's gonna be the best of times in Israel so obviously I think it is a very good thing.