Feet on the middle of the pedals I am 1,85m tall, have shoes sized 58-59 (European) and have classic non sloped frame with seat tube 55cm. Problem is that my feet land with their middle on the pedals, not with the front part where the toes are.

I will try to move seat more to the back, but other than that, is it possible to look for different geometry on the frame, or are almost all frames the same in that respect?
If I understand the problem correctly, it sounds like you're sitting too far forward on this bike. Whether you're too tall for your frame or not is difficult to diagnose without at least pictures, but you might consider that this bike is simply too small for you. (If you're having trouble getting the handlebars far enough forward, that's a sign that this could be the case.) Are you able to get the saddle high enough so your knees are locked with the heel of your foot on the pedal? 

You might also consider simply moving your saddle back on the rails. That might give you a little more room. 

I'd also repeat Wadelp's suggestion - toe clips will help you keep your feet correctly positioned on the pedals.