The timer under a city name is for you and lets you know how much time un till you can help that city again. You can visit another players city as many times as you want daily, but you can only provide neighborly help once every 24 hours
To tell if a city is abandoned you can look up the player on Elvenstats. There are a few visual clues also like if they have a building that is in permanent upgrade mode like their main hall where every time you go to their city it is still upgrading. Some players write in their city banner that they quit playing, some times you can just look at a new city and see that every time you go to it it is not grown any
Don't really know about a grayed out fellowship flag, but an archmage can edit / create their own fellowship banner maybe when they made theirs they were in a gray mood