In fact, it's the use of regional languages like Sundanese which is diminishing. I could really feel the skill and the vocabulary range of even the native Sundanese are worse than 10-20 years ago. Lots of mixing with Indonesian vocabulary without knowing the Sundanese words.
It's very different if your native language is only spoken by a small number of people and / or isn't very prestigious.
Then there is, like you said, the capital, Jakarta, where non-Indonesian languages are only spoken at home (or not at all).
And I know for a fact there are is at least one region - but there's probably more - where the local language has pretty much died out. Minahasans, from Northen Sulawesi, no longer speak Minahasan languages, they speak Indonesian.
Malay and Bahasa are 90% the same language. Bahasa is considered a traders' language, much like Swahili - a lingua franca that allowed commerce to happen for centuries.
Source: studied bahasa for two years and used to live on Java - amazing, complex place.
They are speaking Indonesian as a second native language, they will acquire Javanese and Indonesian at the same time but use Javanese in conversation and Indonesian when consuming TV, books and in school. So their grasp of formal Indonesian is going to depend mostly on education level.
As for colloquial Indonesian, due to the internet new colloquialisms spread much more rapidly to other parts of Indonesia. So it probably depends on how old said Yogyakartan is.
No, they speak daily Indonesian. If an Indonesian spoke formal Indonesian it would be comparable to an English-speaker speaking lawyerese (imagine someone sounding like a EULA).
source: 2010 census http://sp2010.bps.go.id/files/ebook/kewarganegaraan%20pendud...