No, you'll be able to do any of the routes in any order (although the recommended order will be Randal->Heath->Markus) and as any House. However, some special content will be locked to specific Houses.
Regarding tattoos: Not planned at this time because of the amount of work tattoos put on artists (all those details!), and the consideration for MC's appearance in CGs. It may be feasible with a Kickstarter stretch goal, though, if enough people would like to see it!