When cutscene mode is enabled, then no, that's kinda the whole point of cutscene mode. When it's not enabled, on other hand, you can, but only to some extent since the HUD (selection panels etc.) is hidden. You can give gather and attack orders, for example, but not train units or research techs. Admittedly, this is a limitation, but it's the best solution I managed come up with, the in-game HUD itself is already pretty cramped in my opinion.
Yeah, as shown in the second half of the demo video, this is possible. Currently, the unit's displayed name (e.g. "Champion Infantry Pikeman") is used in combination with its in-game portrait. Would be possible, though, to extend it to allow customising those too.