Not true. Most Catholic hospitals employ chaplains for their Catholic patients who are not priests. There are large national organizations approved by the USCCB like this. So, no, hospital chaplaincy is not provided solely by priests and neither are all hospital chaplains priests.
Catholic Encyclopedia says that a “chaplain” is a priest. If the term has expanded in its meaning in general usage, that would create a confusion.
The Sister (or a lay person, etc.) can do a great deal for patients, but can’t offer the Sacraments, of which Reconciliation and Anointing are particularly relevant to hospital patients.
I doubt that.
Can. 564 A chaplain is a priest to whom is entrusted in a stable manner the pastoral care, at least in part, of some community or special group of Christ's faithful, to be exercised in accordance with universal and particular law.(emphasis added)