Dear Knightmare,
Breaking your questions down -
Q1 - should you be sole executor or be co-executors of your brother’s estate -
If you are worried about your other brother being unable to handle the responsibility, you should definitely be one of the executor. As to taking over as sole executor, there is no right or wrong answer - if sole, it would place the responsibility squarely on you and there could be emotional and legal concerns because if you did not do the job properly, if significant enough you could be personally liable. But if you are co-executors and there is disagreement, it could delay and if serious enough may have to resolve dispute in court, which would be expensive.
Q2 can you claim your brother’s estate in entirety including parents portion and the portion of the dispised other brother - if there is no will, the law of intestacy would dictate your share. If you believe there is evidence of your brother’s intention outside of a will, you need to seek legal advice if it is worth making an application to the court to adjust the your estate’s entitlement.
Q3 what do you mean by “cascading intestate”? This is not a commonly used phrase in Australian estate law - please note if this is not within NSW, the situation may be very different depending on the law of that state/country.
Good luck!
-Nighthelyn