The situation is pretty much identical across the Knanaites, regardless of whether they are OO or OC. They are a strictly endogamous community and claim to trace their origins to 72 families of Christian Jews, comprising about 400 persons, who emigrated to India in three ships about 345 AD under the leadership of Knaithomman or Thomas the Cananite. The immigrants are said to have been accompanied by a bishop, Mar Yausef (Joseph), four presbyters, and deacons.
The vast majority of them are found the in either the Syro-Malabar Catholic Church or the Syrian Orthodox Jacobite (Indian) Church - each of which has a formal canonical jurisdiction for them (erected 1911 and 1910, respectively, if I remember correctly), but there are also smaller communities within the other Orthodox and Catholic ecclesial communities in India (except among the Latin Catholics) and, officially or otherwise, provisions are made to accommodate their praxis in all of those Churches.
As you surmised, marriage contracted outside the Knanaite community will effectively result in one's ecclesial ties to the community being severed, although the individual isn't excommunicated. Crossing boundaries between the OO and OC Knanaya communities is acceptable, however, as the bond of the community is apparently stronger than that of the ecclesial affiliation. Marriage across that line will ordinarily result either in the female translating to the male's Church or the Catholic translating to the Church of the Orthodox spouse. In North America, I think the latter is almost always the case; in India, I think the distinction may be more a matter of local praxis.
(There is a Knanaya body within the Marthomite Church - a Protestant church - but both OO and OC Knanaites reject its legitimacy and those who convert to it are lost to their Knanaite community. In that particular instance, my impression is that the degree of separation is almost akin to the shunning practiced within Amish communities.)
Since you referenced reading about the Knanaites at CAF, I'm certain you have heard that there is a great deal of controversy surrounding them at times. From the outside, there is much hostility aimed at what is perceived as an exclusionary body and one the justified existence of which depends on matters often perceived as fairy-tales or recently invented claims to origins in antiquity.
From within, the controversy is relative to the endogamy and the distinct separation that they attempt to maintain, even from their co-religionists (in large part to lessen the likelihood of intermingling between their young folk and other Indian Christians, with resultant risk that the former will contract marriages outside the community). Although they won't bar non-Knanaites from worship with them - and they will worship with their co-religionists if they are located somewhere in which there is no Knanaite temple, they much prefer to worship with their own and be served only by priests of their tradition. Obviously, the difficulties involved in maintaining such tightly knit communities increase when their members relocate to the West - particularly the US, Canada, and, I suppose, Australia. (Although I don't know anything about their numbers or situation in Oz, I suspect it's pretty similar to that here.).
In North America, I think that the issues are currently festering mainly on the OC side, but are beginning to make inroads to their OO counterparts as well. The OO, I believe, have provided a separate diocesan structure for the Knanaya in the US/Canada; the OC have not. The OC Knanaites have only a deanery within the Eparchy of St Thomas in Chicago of the Syro-Malabars and, from what I understand, they're not pleased with that nor with the fact that outside the personal jurisdiction established for them in their historical homeland (the autonomous Metropolitan Arch-Eparchy of Kottayam of the Knanaites), they are not necessarily assured clergy of their own tradition, having to sometimes 'make do' with Syro-Malabarese presbyters. The biggest issue to them in regard to that situation has nothing to do with liturgical praxis or anything of the like, it's a (probably correctly) perceived lack of support from those clergy for the community's insistence on maintaining absolute endogamy. The OCs believe the situation would change had they their own hierarch.
However, there are overt moves among some younger folk (and some not so young) in both the OC and OO communities in North America to break free of the bonds of absolute endogamy. You have to understand that the community stance is so iron-clad that, as an example, they won't accept adopted children of married Knanaites, unless the children were themselves born of Knanaite parents.
Here's three promises
from a 2009 election campaign for office in one of the North American Knanaite organizations (this one happens to be an OC membership group):
4. KCCNA will stand with our youth for their right to choose their life partner without fear of loosing membership in the community. However, KCCNA will encourage our youth to choose their life partner from our own community.
5. KCCNA will stop discriminating against adopted children of Knanaya families and proudly welcome them to our next convention.
7. KCCNA will sponsor a DNA test campaign for our community members so that we can prove our purity and justify our practice. If the test results prove that we are a pure community, we can proudly continue our strict ENDOGAMY practice. In case the test shows that we do not have a pure Jewish background, we can stop expelling our own people and their families from our community in the name of purity.
(emphasis in original)
Note that these promises are being made by the same candidate - I promise to let youth marry anyone and you can bring the mongrel children you adopted to the next convention but we'll do a DNA test and if we are pure, we'll continue strict endogamy. Oh, ok, no contradictions there
So, that's a thumbnail version of what Knanaites are all about. I stand to be corrected on any of this by my good friend, Deacon Phil, but I'm pretty comfortable suggesting to you that with the expected differences in faith and the usual - "who's commemorated in the diptychs?" - what's culturally true of OC Knanaites is true of OO Knanaites and vice-versa.
To your specific questions posted above.
They would be considered members of the larger Church within which their Knanaya community existed - so, for most Orthodox Knanaites, the Syrian Orthodox Jacobite (Indian) Church, under the authority of whose local bishop they will be. Their children will ordinarily not receive the Mysteries of Initiation in a Knanaite church, nor be registered there (the parish metrical books are basically the record on which the community relies to document its continued endogamous purity). They can attend and worship there and be communed there, but cannot hold office, as sheenj has said.
But,, the possibility always exists that they may not be especially welcome to worship there - some communities may make it uncomfortable to do so. Also, an Orthodox Knanaite who has effectively ceased to be such would also be considered as not one were he to seek acceptance to a Knanaite Catholic community - and vice-versa.