Is it my commander?
To reiterate what is stated in the question, your commander is always your commander, and nothing can change that. This is because of the following rule:
903.3. [...] This [commander] designation it is an attribute of the card itself. The card retains this designation even when it changes zones.
Does a commander deal commander damage?
Even if it's morphed?
Yes. Commanderness cannot be hidden. It's an attribute of a card, not a card face. This is specifically covered by an example under rule 903.3.
A commander that’s been turned face down (due to Ixidron’s effect, for example) is still a commander.
Even if it's morphed from my hand?
Yes. Being unable to immediately prove* commanderness is not a legitimate argument for ignoring the rules. To illustrate this very clearly, consider these two rules:
- Correct rule 903.3
- The commander designation is an attribute of the card itself.
- Incorrect rule 903.3
- The commander designation an attribute of the card itself if and only if you can prove what card it is.
*Note that you will eventually be required to reveal your morph via comprehensive rule 707.9.
Am I required to immediately tell the truth if asked if a morph is my commander?
It's not clear. The EDH Rules Committee previously stated on their forum:
Face Down Commanders
Since the card itself is designated as the commander, face down commanders still deal commander damage. Your opponents are also allowed to know which morph creature is your commander.
MTGCommander.net is cited in the comprehensive rules as being a source of "additional resources", so rulings made by the EDH Rules Committee are pseudo-authoritative. However, the forum has since been taken down, and with it the ruling. For lack of an authoritative source, it is unclear if "commanderness" is free information.