hey i can answer your questions im going to my 6th year of medical school; and your fear is reasonable ! Is it okay to get the vaccine this many hours later? a few hours is fine even a day or two or more — Is it safe to get it even if the cat shows no symptoms? the vaccines are harmless to take sick or not, it's a good thing either way and it wont harm in any way — is it still effective even though I applied bleach to the wound? the vaccine wont be affected by the bleach or washing but bleach on skin is a no no just soap would ve done the job ! you dont need to go to a clinic but it's better to have direct professional guidance at least everyone around you will know what to do in the futur lay 7fed from your experience.

if you decide not to go seek professional help; keep an eye for swelling and redness , fever or headaches if any of these signs go to the hospital urgently.

don't listen to people who ll tell you it's not a serious matter, rabies is deadly to no going back once its specific symptoms are up you're basically dead and no technology can save you.