Not all USB-C hubs work well with iPadOS. When choosing a USB-C hub for your iPad, you should select a hub that has specifications that explicitly list compatibility with iPadOS. You also need to consider your power-budget; the hub also requires power for its electronics - and as such you will require a higher-output Power Adapter…
Most models of iPad require at least 12W to reliably charge; a typical USB-C Power Adapter will require ~10W to power its electronics. As such, assuming that you are using a standard Apple 20W USB-C Power Adapter, there may be insufficient USB PD pass-through power to charge the connected iPad.
As for your second question, as all recent models of iPad support connection to TV or external monitor using a suitable HDMI or VGA Adapter - I presume that you actually refer to the new iPadOS16 Stage Manager feature.
Due to limitations of internal hardware, the new Stage Manager feature, introduced with iPadOS 16.1, is only available for these models of iPad:
- iPad Air (5th generation)
- iPad Pro 11-inch (all generations)
- iPad Pro 12.9-inch (3rd generation and later)
Feature availability is usually limited by system hardware - CPU and available RAM being critical to some advanced functionality.
iPad models supporting Stage Manager have an M2, M1 or A12X CPU (upon which the M-series CPUs are reportedly based) - with at least 8GB RAM. The iPad Air4 has only 4GB RAM.
Whether or not a functionally limited version of Stage Manager, compatible with models of iPad that do not meet the current hardware requirements, will ever be released is unknown.