Having researched Civil Service, I'm still not allowed to build a Farm adjacent to an existing foreign Farm, even though I do have an Open Borders agreement with that civ. Is is because the foreign civ hasn't researched Civil Service yet, therefore its Farm doesn't carry irrigation yet, or it is generally not possible for water to be carries from a foreign Farm?

Your first assumption is correct. Researching civil service means that your farms will carry irrigation, but not that they will be able to benefit from the irrigation of adjecent farms. This can be to your advantage as well, if an AI has researched CS and you haven't, since you will be able to use their irrigation to build otherwise unbuildable farms. :)

Also note that farms in neutral territory never counts as having discovered CS no matter who build them originally.