In a LAN network with multiple Access Points, if one of them is awful, broken or old, will the others, shiny new and happy, be hurt? Will the entire network be affected somehow by that bad AP?
Access points don't "wear out." Generally speaking, they either work or they don't.
That being said, if your old AP can't run newer protocols, it will keep others from operation at higher data rates. A 802.11b AP will prevent newer APs from operating at g or n speeds.
Access points can (and will) interfere with each other if they are on the same channel and within range of each other. A rule of thumb is to keep the signal level from an adjacent access point below -85 dBm within the cell area of the neighboring AP on the same channel.