Did you happen to test your "brand, new, it's good" starter before installing it? The test itself is just taking a car battery and a pair of jumper cables, attaching the positive side to the positive terminal on the battery and the 3/8ths copper stud on the solenoid, and then attaching the negative to its battery terminal and then just touching the other end to the case. If good the starter should wind up and motor like crazy. Heavy, heavy whir.
IN situ, do you have all the wires that used to be attached to that heavy copper stud and the nut tight? is that skinny wire that goes to one of the #10 studs, think the inboard one nearest the block, attached to the correct stud and the nut tight?
Did you take some time to put that new battery on a charger and bring it right up to full boogie? A dealer can say that his batteries are fully charged but they lose zots just sitting on the shelf and don't get shipped fully charged to begin with. It usually takes a day, depending on the charger to get the battery to a full load.
El Cam's comment about the battery cables is valid. Even though they may appear sound, they can be severely corroded and nasty internally and you would never discover it unless you pealed the insulation away. Easier to score a new pair and have done with it.
Not sure about that alarm. It served your Mom well, I would guess, but they are like most things mechanical and can die of old age and rot and rust. Might be wise to decommission it completely and remove it from whatever circuit it has been tied to.