Short answer: (1) It is in most cases done by postcards attached to the physical magazine. It generally costs ～$3 to cast a vote (magazine+postal cost). These days, apparently there are polls online. I suppose issues are handled individually, but it is possible to cheat.
(2) Regarding the list, it is simply decided by authors or editors, but in some cases, the list is open and voters can just write their favorite character's name.
This poll (2021) seems to have been collected online whereas this poll (2022) clearly states it is votable only by postcards as follows (the bold part):
This wikipage has a bunch of episodes. Just randomly translated a few:
- In a Nisekoi poll, there was a person who sent 1500 votes to a character. This means the person spent a few thousand dollars for this vote(s).
- In another Nisekoi poll, this voter was voted and ranked 19th. (Here the list was open.)
- In a Bleach poll, there were votes by cardboard boxes full of votes (which I assume are for the same character), but these were counted as just 1 vote, because the magazine side decided it was fairer.
(a side note: Nisekoi was a Shone Jump comic. Shonen Jump is about the size of a large university text book 257mm x 182mm, a few cm thick. Just think of 1500 copies of them...)