First of all, very few companies can afford a full time programmer solely for PLC programming.
Also, the answers may differ depending on whether they apply to machinery manufacturer or end user company.
I am working for a packaging machinery manufacturer, and can share my 20 years experience.
... is your job limited to solely programming a PLC or do you get involved with programming robots, smart cameras, and etc?
Internally to our company, I program everything programmable, that is PLC, HMI, robots, vision, motor drives, intelligent sensors, certain PC applications, etc.
In other words, everything necessary for the machine functionality.
We do not program simple integrated equipment, e.g. printers, because they are typically being started up by end user.
Some projects require third party machinery control reprogramming on customer site.
We do not program the user side of networks not involved in the machine functionality, e.g. SCADA.
Are you primarily comfortable with programming PLCs, or are you comfortable with the other technological aspects involved in maintaining and connecting the PLC via ethernet, devicenet, and etc?
No OR here. I am primarily comfortable with programming PLCs, AND comfortable with the other technological aspects we use?
Is it common for non-technologically inclined people (i.e. managers) to work with the PLC, or is it strictly limited to people specially trained to use them?
No it's not common at all, if you mean system design or PLC programming.
Non-technologically inclined people are not capable of this.