1) That is probably a question you would have to ask the public library about. If you happen to work at the public library and this machine is dedicated to your use then it might be okay (with the library) to install your own software on it, though many businesses require their control of what software is installed on their machines.
2) Your subscription cannot be shared. If this is a machine that is accessible to anyone other than yourself then in order to be in compliance with your license/subscription agreement, you would have to sign in and out of it so that it is only available for your use.