Explanation
Backward Compatibility means that the Management Server is able to manage older Gateways. The lowest supported version is documented in the Release Notes of each version. The Installation and Upgrade Guide only provides information about how to install or upgrade the Management Server and the Gateways, not about the compatibility between them. References: Check Point R81 Release Notes, page 6.

Check Point's FW Monitor is a powerful built-in tool for capturing network traffic at the packet level. The FW Monitor utility captures network packets at multiple capture points along the FireWall inspection chains. These captured packets can be inspected later using the WireShark.

Explanation
The command to identify the NIC driver before considering about the employment of the Multi-Queue feature is ethtool -i eth0, where eth0 is the name of the network interface. This command displays the information about the driver and firmware version of the NIC, as well as other details such as bus-info and supported features1. The Multi-Queue feature requires a NIC driver that supports multiple transmit and receive queues2.
References: : ethtool(8) - Linux man page : How To Configure Multi-Queue NICs | Linode Docs

Explanation
The utility that allows you to configure the DHCP service on Gaia from the command line is sysconfig. This utility provides a menu-based interface for configuring various system settings, including network interfaces, routing, DNS, NTP, SNMP, SSH, etc. One of the options in sysconfig is DHCP Server Configuration, which allows you to enable or disable the DHCP service, define DHCP scopes, set lease time, etc. References: Gaia Administration Guide R81, page 29.

Check Point API is a set of web services that enable the usage of functions and commands in a dynamic and automated fashion. Check Point API is available in different types, each serving a different purpose and functionality. According to the Check Point Resource Library1, the following are the types of Check Point API available in R81.x:
Identity Awareness Web Services: This type of API allows external applications to send identity and location information to the Security Gateway, which can then use this information for policy enforcement. Identity Awareness Web Services can be used for scenarios such as guest registration, captive portal, identity agents, etc.
OPSEC SDK: This type of API provides a framework for developing applications that interact with Check Point products using the OPSEC (Open Platform for Security) protocol. OPSEC SDK can be used for scenarios such as log export, event management, anti-virus integration, etc.
Management: This type of API allows external applications to perform management operations on the Check Point Management server using RESTful web services. Management API can be used for scenarios such as policy installation, object creation, configuration backup, etc.
Mobile Access is not a type of Check Point API, but rather a feature that provides secure remote access to corporate resources from various devices. Mobile Access uses SSL VPN technology and supports different authentication methods and access scenarios.