Regardless of your Mom's goals, the legal thing to do is to follow the laws of intestacy in New Jersey (or maybe PA, depending on whether you consider him to have moved to NJ or if he was merely visiting when he died). This probably means that your brother's estate will pass to his siblings and/or his parents in some predetermined ratios. The girlfriend is out of luck, although you could give her something afterwards if anyone wanted to.
His estate will pay federal estate taxes of zero, because he is under the federal exemption amount of $5.x million. It appears that his estate would owe NJ inheritance taxes on the portion of his estate that his siblings (but not his parents) receive. The taxes would be based on the amount transferred to his siblings, regardless of where the siblings live or their ages. It looks like the inheritance tax would be 40% of the amount transferred minus a $500 exemption.
Generally speaking, when an heir receives something from an estate, they do not owe any taxes on it - the estate already paid the taxes due. In particular, it is not treated as income and you don't owe income taxes on it when you receive it. Of course, once something is in your name, then you would owe taxes on it like normal.
As far as settling the hospital bills, it seems that he had enough money to pay them. Assuming they are legitimate, the reasonable thing would be to pay them out of the estate's assets. You may look into whether money can be removed from the IRA and/or 401k without penalty given the circumstances; I believe there are exemptions from penalties for medical expenses. That all being said, if you call the hospitals/doctors and ask what they would take to settle it all today, they'll probably give you a discount of (just guessing) 10-25%.
Your mom, as the executor, is obligated to pay the expenses of the estate - including any final medical expenses, burial expenses, debts, and taxes - before distributing anything to the heirs. If she takes and/or spends any of the money from the estate before paying the estate's debts, she could be in serious trouble.
Your Mom can't just willy-nilly give money to people from the estate. It is not her money to give; it belongs to your brother's estate. Once you have gone through the inheritance process and the money becomes the heirs, then yes, they can give money to anyone they want, up to $14,000 per person per year tax free.
Until the estate is settled, she should be filing estate tax returns. The estate pays taxes like an individual although the brackets are smaller, which means that it is in everyone's best interest to get the estate settled sooner rather than later because the income earned in his accounts is taxed more aggressively than it would be in your names.
Finally, once everyone inherits their portion of the IRA, they will be obligated to start taking RMDs from their inherited IRA. Consult the IRS for the amount, but the basic idea is you take the amount of your portion of the account as of the end of the previous year and divide it by your life expectancy in years. You have to take at least that much every year. And that RMD, assuming your brother had a traditional IRA, counts as taxable income to you.
It's a complicated subject, but hopefully some of that helps.
Good luck!