Thus I think it will be impossible to re-select the ICA components to discard. Thus, should I perform again the ICA process (taking the signals before the old ICA) rather than add another layer of ICA?
Re-using old file structures that you are not sure how to manipulate is risky.
If you want to update your pre-processing pipeline, it's probably safer to start over from continuous files.
In that case, compute the ICA decomposition with runica from Brainstorm.
For example, in the link you share it use VEOG to detect the blink components. Is it possible to use HEOG?
VEOG is best for detecting blinks.
HEOG is best for detecting saccades.
https://neuroimage.usc.edu/brainstorm/Tutorials/BadSegments#Saccade_SSP
It is relevant to select a right or left EOG channel?
EOG should be used as bipolar channels (eg. EOGL - EOGR).