Does scratching your eyes affect your allergies?

No. Allergies can cause itchy eyes. Scratching your eyes is not a good idea, as it may cause infection, and you could also scratch your cornea. For the same reasons, you should not wear contact lenses if your eyes are affected by allergies. You could use oral antihistamines like zyrtec (cetirizine) and/or antihistamine eye drops like Pataday to control the itching.
Yes. If you are rubbing your eyes, you are potentially exposing your eyes to additional allergens worsening the symptoms. Additionally you risk other issues such as corneal abrasions and eye infections.