Berlin

Weather north of the Alps is iffy, period. It might be a lot warmer than weather averages would lead you to expect. Or it might me quite chilly and wet.

Go to wunderground.com for lots of historical weather data on the places you're asking about. You can get day-by-day actual stats for the last 10+ years by choosing History> Monthly.

Room rates vary by week and day, not just by month, so you need to do research for your specific dates on a site like booking.com.

I think Rick lists a lot of the major special events right on this website (probably in the Explore Europe section), and of course they will also be mentioned in your guidebook(s). You need a book or two.

I have one trip's experience on the weather in Berlin in April. That was in 2014 when I was in Berlin the last week in April, left on 1 May for Vienna. I found the weather to be still cold.

If you're looking for the month when the cheapest room rates are offered in Germany, (and in Wien too), if at all, by the hotel/Pension, then that would be in late July to the end of August, ie, what we call peak season, but in Germany it is seen the "slow season."