Can the pinyin for 边 be either "biān" or "bian" in general?
As I answered in another question you are referring to, those 边s should be pronounced as the first tone biān in standard mandarin except when 边 connotes 'positions' and 'directions', such as 里边 前边 东边 左边 这边 and 那边 in your list. Other than that, the rest are all first tone biān. 

However, in spoken or daily usage, lots of people often pronounced it as neutral tone even for those that should be pronounced as first tone as dictionaries indicate. 

The best way to get it correct all the time is to look up any authoritative dictionaries, such as 新华字典, 现代汉语词典 and etc. Some on-line dictionaries might not be right all the time because they are not well proofread. Usually, they just copy here and there. 

Updates:

After I looked up in the latest dictionary 现代汉语词典， I have some updates on this answer above accordingly.