1) Yes, that's done with the Choice Settings of the GUI box. You need to set the number of Choice Columns you want to have and set the Column Fill to Horizontal (column fill is only available when more than 1 column is set).
2) Images can be used, e.g. just set up an icon for that choice, or use text codes to add an icon/image. Different image if it's selected is not possible, though.
3) No.
4) No.
Also, a log menu doesn't really allow you to choose anything and just displays information - so you can't really have any character selection results from that.
Something you could do would be setting up an event with a series of dialogues. Based on what you selected in a dialogue, you could display it again with the previously selected choice using a different image (#2) and having it on top of the list (#3). Displaying the info for that selected choice can be done by showing another dialogue (before showing the choice dialogue again), using type Message and disabling Wait.
Make sure to use a Close All Dialogues node when selecting a different choice (and at the end of the event) to close the dialogues that are kept open.