SSI and SSDI are two seperate disability insurances. As far as I'm aware, you cannot receive both, although both are usually applied for at the beginning of the determination process, just as a matter of efficiency.
SSI is awarded to those who have not worked in the workforce enough to have contributed (through paycheck deductions) to their Social Security. The SSDI is awarded to those who have worked and contributed toward their Social Security by paycheck tax deductions. The SSDI will usually be the larger amount awarded of the two.
That being said, and assuming that you have not worked for a sufficient amount of time to qualify for the SSDI, and it is later determined that you are competent to run your own financial affairs, you should be able to draw SSI under your own name eventually.
I draw SSDI and have a young son, who also receives an amount under my SSDI. Since he is my own child, and not a foster child, etc., I do not have to supply receipts or itemize how his money is spent, but I do have to fill out a yearly account of where this money goes, in general, aggregate totals. Any accounts I keep for him must be in his name, listing myself as the Payee.
Now, I cannot speak to how SSI requires accountability, but I'm assuming, since you are receiving money under your mother's name, that once you have been awarded SSI payments under your own name, that your child will also receive an amount under your name, just as you are now receiving monies via your mother's SSI.
You didn't mention whether your mother is also receiving SSI for herself as well, so that may come into play as far as how the benefits are computed and handled....but definitely check it out, because I only know how SSDI is handled, and I'm really just assuming here regarding the SSI.
Also I would suggest calling your rep. and inquiring to make sure that your daughter is receiving whatever she may be currently eligible for, as well as what she will be eligible for once your SSI is initiated in your name.
I hope some of this helps. Do go to the Social Security site and read there....it is very helpful and informative.
About the marriage question...with SSDI, when I married, it did NOT change the amount of my monthly award, but it did create a situation whereby a portion/percentage of my SSDI is taxed at the end of each year. This taxable amount is minimal and most of my yearly SSDI amounts are not taxed.