Ignore 'Curtisd'... he is a result of a partial abortion.... they only got his brain!
In answer to your questions:
Q1) "My brother was arrested for dui, after the cops arrived at our house 10 mins. after he arrived home after hitting a mail box."
A1) I assume that the was evidence left at the scene or a witness that was able to lead the police to your address. As such, the 10 minute delay is of no relevance.
Q2) "The cops arrived to investigate the hit and run asked my brother to come out side and proceeded to do sobrity tests. While doing the tests the cops discovered empty beer bottles in to back end of the truck while looking at the damage. After the tests he was told he was to he was under arrest for dui but was not read his rights even after he was already asked questions."
A2) They should have read his rights prior to asking questions.
Q3) "They finally read him his rights twenty minutes later at the station and asked the questions again."
A3) They obviously realized their error and corrected it by making sure he was Mirandized and then requestioned. By doing this, they negated any impact from the first set of questions.
Q3) "Can you get dui if they did not see you driving and asked you to come out of your home?"
A3) If there is evidence that he was driving (by witness or admission) and he flunked a field sobriety only 10 minutes after arriving, then yes.
Q4) "Can they search the back of the truck without consent?"
A4) I assume you are talking about the bed of the truck. If so, the 'evidence' was in plain sight and is subject to search.
Q5) "Are you supposed to be read your rights immediately after you are informed you are being arrested or can they do it later at the station?"
A5) Though normally prior to questioning (and if not, the answers could be inadmissable), there are no requirements as to where the Miranda rights are set. And in this case, the fact that they did Mirandize and re-asked the questions, removes this as an issue.