First, you need to update CPUSE on the management. This can be automatic, or via thumb drive or whatever for an air-gapped management server. When you do any action with CDT, if the management's version of CPUSE is newer than the firewall's, CDT pushes the management's version to the firewall and installs it.
You don't need Internet access from any device. All you need is a way to get the jumbo you want to install into CDT's repository. This requirement can be fulfilled by giving the management access to the Internet and having it download the jumbo, or you can download the jumbo on some other system and bring it to a system with SmartConsole via thumb drive, DVD, or whatever.
Once the jumbo is in the management's repository, the management can push it to the firewalls. If the firewalls have access to the Internet, the management can also tell the firewalls to download it from Check Point themselves (nice for firewalls which have a big Internet connection but which are managed over a smaller WAN link). When you start the patch process via SmartConsole, this is controlled by the "Package Location Source" picker.
As for your second question, yes, the jumbo is installed on all standby members of the cluster first. Once the first standby is back up and talking with the management and the cluster reports it's healthy and synchronized, the cluster is failed over and the jumbo is installed on the formerly-active member. Upgrades follow a similar process, just with 'cphaconf mvc on' to enable cross-version sync before the failover.
The admin takes a single action to start the process, and that's it. All of the update or upgrade steps are handled fully mechanically.
Note that the cluster needs to be healthy. If any of the cluster members have any status other than ACTIVE, STANDBY, or BACKUP, the failover may not work.