What are the key points organizations must consider when interpreter services are provided by staff and practitioners ?

Any examples are for illustrative purposes only.

Practitioners (nurse, physician, etc) communicating directly with a patient in their preferred language while providing care without the presence of an interpreter

There are no standards that prohibit a bilingual practitioner from communicating directly with a patient in another language while providing care, treatment or services. However, it is recommended that the organization has a process to make sure that communication with the patient in the non-English language is effective and meets the patient's needs. For example, the organization can determine if a language proficiency assessment is necessary to make sure the bilingual provider is able to communicate effectively or may consider using an interpreter to validate the patient's understanding of the information provided by the bilingual provider and communicate further information as needed.
 
Additional Resources:
The American Medical Association's Promoting Appropriate Use of Physicians' Non-English Language Skills in Clinical Care: A White Paper of the Commission to End Health Care Disparities provides guidance for physicians on the appropriate use of their non-English language skills when caring for patients with limited English proficiency.

Applicability of HR.01.01.01 to providers (nurse, physician, etc) serving as an interpreter for a fellow provider

The requirement found at HR.01.01.01 'Note 4' applies to bilingual providers serving as an interpreter to bridge the communication between a fellow provider and a patient. For example, the patient and his/her provider speak different languages and a bilingual provider is facilitating communication between them. 

Organizations surveyed under the Behavioral Health Care (BHC) manual
RI.01.01.03 requires that individuals providing interpretative services are trained to provide such services.  Examples may include:

These options may be provided in person or via telephone or video. The documents that are translated, and the languages into which they are translated, are dependent on the population(s) served by the organization.