Actually the Maronites are the one Eastern church that was never separated from the Catholic Church and the pope. As such, there are no "Maronite Orthodox" churches. Fr. Mitch Pacwa on EWTN is bi-ritual with Latin rite and Maronite rite.
Some of the Eastern churches attempt to prevent "Latinization" meaning preserving their own traditions and culture rather than being influenced by the Latin rite. The Maronites are not like that and in fact, other Eastern churches criticize them for allowing their church to become overly "Latinized". Well my opinion is why can't all the different rites influence each other? Why can't Latin rite churches have more of the beautiful religious icons that we see in the east, for example?
For example I saw a Maronite priest wearing a badge of the Sacred Heart. Some Eastern Catholics would say "that was private revelation that came in the Latin rite so it should be confined to Latin rite churches". Well I think that reasoning does not make sense since if God sends us a message of His love he means it to be for the whole church. They say the same thing about the Rosary, the Divine Mercy devotion, etc.
The Maronites, in my experience, do not tend to be like that and have allowed themselves to be very influenced by the Latin rite, perhaps too much so as to lose some of their own cultural identity. But I think some sharing across the different groups is healthy.
The Melkites you will find on the other hand tend to stick very much to traditions that come from the East and Orthodox churches, and try to avoid any hint of Latinization. They actually have some different beliefs that are more similar to the Orthodox beliefs, but are still in union with Rome. The Orthodox church from which they came is the Antiochan Orthodox. If you want to learn more about the Melkites, check out this site:
http://www.melkite.org/. It is the site for the American eparchy (aka diocese) of Melkite churches, and explains a lot about their beliefs and practices.
Lastly, regarding religious communities, I know of some Maronite monks in the US that accept members of other rites. Their site
here says "Those wishing to join this Eastern Catholic monastic community are not obliged to become members of the Maronite Catholic Church. If one was baptized Melkite or Latin, for example, he remains Melkite or Latin." So with this wording it sounds like it may be up to each community to decide.