Lets take your questions from the bottom and move up . . .
Some companies have their own proprietary websites, many rent space on other platforms. The most frequent platforms are Sassie, Prophet and Shopmetrics, but there are others. In a few cases several companies are owned by one company. This stuff is not something you really need to be concerned about.
Excel worksheets have always been my best friend for keeping track of information. I have a spreadsheet of all the companies I am registered with that has a column for the company name, a column for the login page of the website, a column for user name, a column for password, the day I applied for the company. Additional columns are convenient for phone numbers, scheduler email addresses, payment schedule, etc. Excel worksheets are also the way I do my accounting, showing the job date window, the client, the address, the company I'm doing the shop for, information such as the hours during which I can do the shop, the date I did the shop, the invoice date (if an invoice is needed), the fee agreed to, the bonus that may have been agreed to, the amount of the reimbursement, the date I got paid and the amount I got paid, the mileage incurred for the shop.
It has been a long time since I did a movie count and I always found that standing in the back of the theater was the best way to do that. Read your instructions.
The comic book shop is pretty straightforward, again, read your instructions.
The biggest thing with your gift giveaway shop is to make sure you keep the cards absolutely secure and that you get all information from the winner before the card goes into their hands. Returning leftover rewards in years past has been subject to issues. Take a photo of all cards going into the return envelope and note the card numbers. Ask someone else to verify what you are packaging to send back. Send back by some shipping method that is insured and has tracking. I don't know what they do these days as return envelopes, but make sure it is secure.