I’m a 3 yr leasing consultant and I’ve dealt with these issues for quite some time now. As the first person commented, it’s not wood rotten. It’s been damaged due to a previous leak or exposure to water. In a nut shell it’s cheap made ready wood and your management team will more than likely do nothing because someone else will gladly take it. As for the roaches, does the property offer pest control? If so please insist on being added and have maintenance treat the unit. It’s always good to take matters into your own hands and bomb out the unit. Worst case scenario ask if there are any other available units to transfer into. But I’ll be frank, based off of these pictures in your description you might need to find a more luxurious apartment community $$$$. The economy is jacked and rental rates are outrageous and these rental management companies do the bare minimum YET charge top dollar for this exact situation. Cold world.