That really is a weird one, because here in the UK we say 'fortnightly' (for 'every other week'); so 'biweekly' would definitely be taken to mean 'twice per week'.
That also matches 'biannual' (an event that occurs every other year is 'biennial'), so 'biweekly' as 'fortnightly' is certainly to be proscribed, in my book.
I only have Collins' concise to hand, which gives the definitions I have and doesn't even mention (not even proscribed or US etc.) other uses. And Collins is pretty permissive.
And no they're not. 'Biennial' events occur once per two year period; 'biannual' events occur four times per two year (twice per year) period.
For example, in horticulture 'a biennial [plant]' is one that has a single flowering/reproduction cycle over two years. (Cf. 'annuals', 'perennials'.)