When I play free play, my customised character doesn't appear but instead some default one. Is this normal and just another failure?

i too have seen this try dieing once and that usally fixes it u will respawn ur character

Happens a lot when you do multiplayer period from what I've seen. Usually fixed by respawning or using a CP... usually being the key word.

The skin has to load, and it can take a few minutes.

Yea, like stated. Usually right after an event in the mission, or changing of the map you'll be able to play as your char. model. Honestly, it's not too surprising. Less load time if they hand you a default so that you can play.