Is it permissible to document charitable donations, and does that affect their reward? Is that considered a form of reminding (of one's favor), and does the Hadith, "Do not enumerate, lest Allah, the Almighty and Majestic, enumerate against you," apply to it?

Creating an account on electronic charity websites to log in when donating is not considered boasting or showing off, as long as the donor does not intend to seek people's praise. However, if logging in results in the participants knowing about the donation, then you should choose between donating while logged in or as a guest, based on the preference between secret and public charity. Secret charity is usually preferable.

It is better not to record these donations, as doing so implies counting, which may lead to considering them too numerous and then ceasing to donate. This is as explained in the interpretation of the Hadith: "Give, and do not count, lest it be counted against you." Nevertheless, merely recording or counting them does not invalidate the reward of the charity, but there is a fear of what is mentioned in the Hadith: considering them too numerous and then ceasing them.