If you join, for example, the Order of the Hour, can you still get services and quests at the Akatosh Chantry (and vice versa)?

Yes, they are the same guild and share membership/services/rank/etc.

They're not the same guild, they're a Cult and a Knighthood that happen to be dedicated to the same god.

Though somewhere deep in the intricacies of the DF reputation system there's probably some perks for joining an allied cult and knighthood, they're distinct entities and don't "share" ranks, nor is it compulsory to join them both.

For the Knight order I usually went with that of Sentinel or one of the other two main storyline provinces, and whatever Temple fit the activities and/or services most needed by the character.

My dear Lady Zilmar, I believe you are confusing Templar Orders with Knightly Orders. The Order of the Hour that Emerald is referring to is the military arm of the Akatosh Chantry; they are temple knights, not a true knightly order such as the Knights of the Dragon of Daggerfall. A temple and its military order are one and the same. If you join one, you join both. Of course, often locating an actual guildhall of a military temple can be difficult. I'd like to say that only 3 or 4 different ones have been found, and perhaps not all are even included in the game.

Temple knights offer the same services and quests as their respective temple. The only difference is the name. In fact,I'd like to say that if you join the Templars, your actual affiliation listed on your character sheet is that of the Temple. An easy demonstration is in Sentinel City, where a Knights of Iron can be found, who I believe serve Stendarr.

Knightly orders, yes, are completely separate entities

Did check and indeed you are right. I remembered for some reason that the knightly orders as well as being tied to a province were under the sponsorship of a particular deity but they are not.