Yes. Messaging fits in the three parts of speech listed.
Firstly there is the verb to message which is transitive and can be used in all the usual ways, e.g.:
I message people all the time.
Yes, Tony! I'm messaging them now!
I've messaged them, OK? Get off my back. Gah!
So you can see messaging being used in the present continuous sense.
Then there is the use of the present continuous sense as a noun, often referred to as the gerund.
The messaging will take place at 3pm today.
And thirdly there is the use of the present continuous where an adjective would go, which is often referred to as the participle. The example in the question of messaging system is a good example of that. You can see that messaging is modifying system much like red or unlucky would.
As far as I know, all transitive verbs work like this.