I had some past allergy from adult aspirin. Would I have the same problem if I take baby aspirin?

Possibly. If you have had a serious anaphylactic allergic reaction to aspirin, you should avoid all aspirin. This type of allergic reaction involves hypotension, severe respiratory distress, swelling of the larynx, and usually requires intubation and epinephrine. If you had a minor rash or stomach upset with aspirin, you could try the low-dose baby Aspirin dose, with food.
Yes. Aspirin is aspirin. Baby aspirin is just a lower dose, commonly 81 mg compared to 325mg adult. We quit giving aspirin to babies in the 1970's when it was linked to Liver failure/coma/death in kids. The name just stuck. It is now commonly used to decrease platelet sticking/stroke/MI. Do not try this without consulting your doc.