Oh, me, oh, my, one of the perennial topics of online theological warriors.
Byzantine Catholics (called Greek Catholics in Europe and sometimes in America; Byzantine Rite = 'Greek Rite') and other Eastern-rite Catholics have the same doctrines as the Roman Catholic Church of course. Rome claims it's the one true church (as Orthodoxy claims of itself) and with that comes the power to define doctrine. That means if you know of the doctrine, you must accept it.
That said, IMO you can finesse every Roman Catholic doctrine into Byzantine theology except the one about
the scope of the Pope.
Liturgical practice among Greek Catholics (I'm old-fashioned so I use the old term) varies a lot, from just like the Orthodox (Melkites and the tiny Russian Catholic Church, consisting of non-Russian born RCs and converts from Protestantism in America) to something nice like the old Latin Mass in a different language to, in lots of places, something with a very Vatican II feel to it only not nearly as liberal as your local RC parish. (A very few places even have some lay people giving Communion and altar girls.)
Rome wants them to be just like the Orthodox in church and again holding all RC doctrines but expressed in Byzantine terms. If you believe as Rome does about the Pope, it's a perfectly sensible and honourable position.
Over the centuries a lot of them disobeyed that and imitated the Roman Rite in varying degrees, sometimes to try to please the suspicious local Roman Riters, sometimes to spite the local Orthodox. (Ukrainian nationalism, part of the territory of Galicia, not the whole Ukraine: the Ukrainian Catholic Church is a big part of that. A few Russianisms - onion dome, a few icons, married priest, Cyrillic alphabet - to show they're not Polish, but lots of Polishisms - clean-shaven priest, RC devotions, even no iconostasis - to show they're not Russian.) As most here know, that's called
latinisation.
(A certain amount of crossover is normal even with the Orthodox, from certain devotions - the icon of Mary with seven daggers around where her heart would be comes from the Poles - and Russian baroque architecture, Russians' Westernised choral music and icons that look like Western paintings... to parishes of the now-OCA, most of which are descendants of Ruthenian Greek Catholics who switched to the Russians around 1900, having Solemn First Communion for 7-year-olds... to the American Carpatho-Russian Orthodox Church, former Ruthenian Greek Catholics who for years after converting in the 1930s kept many of their traditional latinised practices* - the Patriarch of Constantinople didn't try to change them - and in the hollows of Pennsylvania and Ohio some still do.) *Such as First Communion for the 7-year-olds, priests in Latin cassocks, monsignori and bination (a priest celebrating two Liturgies in one day on the same altar).
Your rank-and-file ethnic Greek Catholic is essentially a Roman Catholic with a different, rather better Mass. Most are fine with being called Roman Catholic, Ukrainian (Ruthenian etc.) Catholic, even Uniate (now a no-no in ecumenical talks); just don't call them Orthodox.
A good number of people in Greek Catholic parishes are Roman Riters who rightly hate what Vatican II did to their rite and found a sturdy refuge while remaining under Rome. Fine folk.
The 'Orthodox in communion with Rome' are a tiny minority of Greek Catholics, mostly converts from the Roman Rite or elsewhere and mostly online. Externally they're what Rome wants but they deny the RC doctrines the Orthodox do, yet they're where they are, which doesn't make sense. A lot of them get fed up after a few years and become Orthodox.
All Eastern Catholics are about only 2% of the Roman Catholic Church.
