First off, the pipe organ is THE instrument of preference for the Roman Catholic Church. Inter alia, the organ is that which represents all of creation in praise of God (organon--Greek)
For all but a very few purposes, the organ is more than sufficient--accompanying a congregation or choir, or by itself in voluntary mode.
The piano is a percussion instrument, which in and of itself is slightly disruptive by virtue of the percussivity. The fact that a flood of mediocre-to-horrible "church" music exists using the piano does NOT change the position of the Church on the subject of the preference for the pipe organ.
Generally, the reason that pianos are used is that the "musician" is incapable of using the organ--another topic altogether.
There are bunches of serious papers written on the topic, but VatII's Doc on Liturgy contains the 'pipe organ is preferred' language (or similar.)
I hear and obey. But (and this is a confession) I hate organ music. Maybe it's because the parish I grew up in had arguably the world's worst organist with the most awful singing voice I have ever heard (I kid you not. When I was still a teenager, a friend attended Mass at my family's parish. The organist sang some solo part of the liturgy. My friend almost choked when she heard him. She turned to me and whispered, "Is that how he really sounds? I thought somebody was playing a prank!")
But back to the organ... Does this mean the great masses composed by Mozart, Beethoven, and Bach are off limits because they include other instruments?
As to your question re: "Novus Ordo Latin" Mass vs. "Old Rite/Tridentine Rite Latin" Mass:
The "Old Rite" or, properly, Tridentine Rite Mass was suppressed by Pope Paul VI in 1965 when he introduced the "Novus Ordo." The N.O. is officially in Latin, but over time, all of it has been translated to the vernacular--English, German, French, etc. However, it can be said in Latin--and some people refer to this as the "Latin Mass."
The "Old Rite"/Tridentine Rite was established in 15XX after Luther's revolution. It MUST be said in Latin--in fact, outside of the homily and the reading of the Epistle and Gospel in the vernacular (which was done from the pulpit immediately before the homily) NO vernacular was allowed--not hymns, nada. Zero. Zip.
After a good deal of prayer, sacrifice, and stationery had been consumed, John Paul II officially allowed the Tridentine Rite to be used again in the Church (for other than private Masses.) More blood and ink has been spilled on the topic of "Old vs. New" on these forums (and a zillion OTHER forums) than for almost any other event in the history of the Church with the exception of Luther's revolution (and its daughter, the French Revolution, with all the concomitant philosophical and theological distortions, heresies, apostasies, etc., etc,.)
That's a VERY short and not-perfectly-precise rundown.