For Nike, the Greek Goddess: Nigh-key
For Nike, the US Athletics clothing company: Nike, rhymes with bike (though I believe the usual American pronunciation is the same as for the Greek Goddess), all my friends bar one say Nigh-key though!
For Addidas, as its German, Addy-das, though all my friends seem to say Uh-deed-us.
Shrewsbury is a difficult one, I think I would probably lean towards Shroez-bree over Shrooz-bree.
Regarding place names, there are some less easy to say ones up here in Scotland. The usual one to catch people out is of course Milngavie (Mull-guy), but then there's Wemyss Bay (Weemz Bay), Kingussie (King-you-see), Kirkcudbright (Cur-coo-bree), Anstruther (traditionally Ain-stir (Ain rhyming with pain)) and my personal favourite Kilncadzow, South Lanarkshire which is pronounced Kill-kay-gay!
A few words I'd be curious to know how others pronounce:
Again/Against - I say both with the gain part as the word gain rather than gen.
Yoghurt - yog-urt (yog rhymes with bog)
Ate - rhymes with bet for me
Nephew - I say nev-you rather than neff-you