About how much does language typology correlate with genetic relationships among languages? For example, should we expect most Sino-Tibetan languages to be isolating, or most Indo-European languages to have a lot of fusional affixes, and so on across language families?
Not very much, and what correlation there is can plausibly be put down to length of common development.
Note for example that while Slavonic and Baltic languages retain much of the inflectional apparatus of PIE (though most nominal inflection has been lost in South Slavonic), English and the Scandinavian languages have lost most of it. They haven't quite got to the point where you'd call them isolating, but they're getting somewhere near it.
The Sino-Tibetan languages are actually a wonderful counterexample. Some are isolating; some, especially the modern Tibetan languages, are agglutinative; and the Kiranti languages are highly fusional. Some are tonal; many aren't, including Newari and several of the Tibetan languages. (Indeed, Newari phonology looks like the phonology of an Indian language: voiced and voiceless aspirated plosives, retroflex and dental consonants, phonemic vowel length, and so on.) Some are SVO, but some, including the Tibetan languages again and also Burmese, are SOV.
It's important to remember that languages don't "know" what family they're in. A feature that's been part of the language for thousands of years is just as likely to stick around, and just as likely to be lost, as a feature that was just borrowed or innovated yesterday. So for instance, yeah, Mandarin and Newari share a common ancestor several thousand years back — but now there's nothing that's constraining them to evolve in the same direction as one another. And so in fact they've diverged tremendously, Mandarin coming to look like a "typical" East Asian language and Newari like a typical South Asian language.
To the extent that genetically related languages are typologically similar, it's a result either of historical intertia or of continued contact. Over the course of 4,000 years, historical intertia only buys you so much continued similarity. So when a language family spreads out over a natural barrier (as the Sino-Tibetan family did) or across a wide area (as Indo-European did), the result is an awful lot of typological diversity among genetically-related languages.
I cannot give an exact number of correlation, but the correlation between genetic relationship and typological features is strong enough to be annoying when one tries to construct an unbiased sample of the languages of the world. Picking just languages at random is not a good idea because a few very large language families (Austronesian, Niger-Kongo and Indo-Germanic) contribute a large bulk of related languages.
It is fruitful to include genetic relations between languages in typological studies, as demonstrated by Levinson, S., Greenhill, S., Gray, R., Dunn, M. (2011). Universal typological dependencies should be detectable in the history of language families. Linguistic Typology, 15(2): 509-534