No, you are not required to notify the cellular carrier, but most installations notify them as a courtesy. They let the cellular carrier know they have adjustable gain which will be adjusted in order not to interfere with the local cell tower. They may also provide contact in case the timing method in the cellular tower needs to be extended for longer distance applications.

Please see the following FCC website for further clarification. 

The IDRPT-9000 is an indoor cellular extender and repeater for CDMA, WCDMA, GSM, EDGE, 3G, 4G, and AMPs. From up to 20 km/25 miles away, the IDRPT-9000 takes a cellular signal from a cell tower(s) and repeats it to a building or structure with limited or no reception indoors.

The IDRPT-9000 also adds web-based access for configuration, monitoring, and troubleshooting via IP.

Currently, 850 MHz and 1900 MHz are available and FCC certified. The 900/1800 MHz models are available without FCC certification.

The unit comes with a 2-year manufacturer warranty on the hardware. This includes defective parts. The unit must be sent in for repair depot service. The repair time is 1 day. If it is received in the morning, it will go out repaired the same day.

• The IDRPT-9000 does not have to be close to the originating cellular signal like other products. It can be up to 30 km/18 miles away. In addition, typical cellular repeaters only repeat the signal a short distance while the IDRPT-9000 can repeat the signal similar to how cell towers perform.
• The IDRPT-9000 has not only manual configuration via simple dip switches, but also provides advanced web-based configuration, monitoring, troubleshooting, and alerts via the Internet or a private IP network.
• Many repeaters limit the number of calls due to how they operate while the IDRPT-9000 can repeat as many calls as the originating tower can handle. 
• The IDRPT-9000 was designed to bring the signal into specific places like warehouses, small buildings, or other structures.
• The IDRPT-9000 has a built-in gain of 70db digital auto power control before antennas, many competitors are much less than this or include antennas to skew the specifications.
• The IDRPT-9000 has adjustable gain to work in unison with the cellular provider's tower, the competitors have fixed gains that can interfere with a cell tower.
• The IDRPT-9000 has a 2-year warranty with a 1-day turnaround service, the competitors only give 1 year or less and require up to 3 months for repairs.
• The IDRPT-9000 is FCC certified unlike other manufacturers and Asian products which are self-certified with misleading statements like "We meet or exceed FCC certifications". The current proposed FCC rulings could be a costly mistake with a competitor's brand.

Yes, since the IDRPT-9000 only cares about the frequency and not the type of service, it will pass all the features that are provided to your cell phone.