Not really. You can tell if you have a bad battery when you start it. I will kind of hesitate to start/slow starting. Your lights can begin to dim also. A bad battery will affect everything. It is a good idea to have it checked though. Most placed like Autozone, Advance Auto Parts, OReilly's will check it for free. Since you did change your alternator you may want to have your battery checked just in case especially since you can do it for free. With your interior lights I would start with the fuses. Look inside of the fuse box and pull each one out. If the link inside is broken inside of the fuse then it is no good.