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There's no such thing as "entering the EU". You can only enter the Schengen area (which includes non-EU countries, such as Switzerland) or the non-Schengen EU countries (UK, Ireland, Bulgaria, Romania, Croatia, Cyprus). Each non-Schengen country has it's own visa regime and doesn't care in the slightest about which passport you've used for other EU countries. So to answer your questions...
is it okay to first enter EU (Bulgaria) with non-biometric passport and Bulgarian blue card and after that enter another EU Schengen country
Yes, it's perfectly okay.
If that is allowed and if then I'll back to Bulgaria from Schengen zone will 90 days counter be stopped?
Yes, it will stop, since Bulgaria is not a Schengen member.
Do I have to inform Schengen zone border control about my Bulgarian blue card in order to not get any entrance problems later?
No, you don't have to inform them about it.