I've lived in Berlin for over 20 years now, DH even longer, DC were born here. So one thing we've not done is a family relocation. However, we have abundant experience of the school system here, and as both DC go to international schools I've seen plenty of other families come and go.
Costs in Berlin are generally much lower than either London or Bavaria (esp. Munich). Rental, food, everything. We're not up to date on rental costs because we have a mortgage. But I would be shocked if you had to spend as much as 1,500 euros for a decent 3-bed terrace house. Possibly more like 1,000 or 1,200. There are obviously fewer houses on the market, as more people live in flats. If you go for a flat, make sure it's one of the gorgeous Altbau properties with huge ceilings. They're lovely to live in if properly renovated. ImmobilienScout24 is OK, but you could also try the small ads section of the local magazine Zitty, which often advertises rentals, or the newspaper Berliner Morgenpost.
Nursery costs are strictly regulated as long as it's a state-subsidised nursery. About 50% of families put their kids in some sort of childcare before age 3, which is very high for Germany. But to do so and be subsidised, you have to prove both parents are working or studying. Once the child is 3 you have a right to at least 5 hours a day subsidised nursery care. Subsidised means it's income dependent - we never paid more than 150 euros per month for 9 hours per day, but I believe the highest contribution for really high earners is around 500 euros. An unsubsidised place would be more like 800 euros.
Would you want an English-German nursery place, or purely German so the DC can learn via immersion?
Once a DC hits 5, or the final year before Year 1, I believe nursery is now essentially completely free (apart from a few euros for lunch costs).
The school system is appalling, but I'm possibly a cynic. There are, however, a number of options for free state international schools which offer bilingual education (some subjects taught in English, others in German). I would totally recommend that, TBH, because the teachers are trained to help your English DC learn German properly - in a German school teachers can often be somewhat narrow-minded about 'foreigners'. There are a few English private schools teaching GCSEs etc, they tend to be between 5,000 and 15,000 euros per year, I recall.
We have known families who've hated it here and who have ultimately gone back to the UK. That's tended to be when a trailing spouse (usually the mother) is effectively forced to follow her husband and has no interest in the local culture. These people tend to have a miserable time of things. If you're both genuinely interested in integrating yourselves and your DC, it can be a brilliant experience. Berlin is an incredibly dynamic place, still the alternative energy from pre-1989 days, extremely tolerant to alternative views and gay lifestyles. It's where Germans from more privileged parts of 'West-Germany' come to shake off their bourgeois roots,. Of course, that now mixes with all the politicians and diplomats in Mitte, so it's a weird but fascinating mix. I would hate to live anywhere else in Germany - I'm just too used to having all this culture on my doorstep. It's VERY different to Geneva in terms of lifestyle, so you really need to decide what kind of vibe you're looking for.
I think you'd be pretty comfortable as a family with a net income of 2,500 euros, but taxes are fairly high so your gross would have to be much higher than that. Child benefit is around 184 euros per child monthly. I honestly don't know what our family income is (not hidden from me or anything - I just throw figures at our tax accountant and he does the rest) but we have enough to do everything you say, give DD2 private music lessons, both DC ride, and if we want to go to the opera or theatre we can easily.
You could check out - cautiously - the website Toytown Germany, and go to the local Berlin site there. It's rather aggressive - even more so than MN at its most viperish -but can contain useful info. There's lots of useful school info on the 'Family Life' section. But hey, when I first arrived there was no Interweb and I just muddled through somehow (admittedly without children).
If you need more info on English-German schools we can PM about it.