I have natural aloe pants, can they relieve inflammation and mild rashes as well as a hydrocortisone?

No. There are several concentrations/strengths of cortisone cream. It is doubtful that aloe plants will have the same anti-inflammatory effect as the usual dosages of Hydrocortisone recommended by doctors.
Not as well. Aloe can be very soothing, but it works entirely different from hydrocortisone, a topical steroid. They can help with the inflammation and rash, but you won't know if it works as well for your skin unless you try.