Your questions cover a lot of ground, and I don't think I can even attempt to address all of it. I don't know very much about the video game industry, but I do know a bit about IT, commercial hardware, software, and contract research.
Let me start by saying that Computer Engineering, as a degree, is kind of an "in between" degree - I have a BS in computer engineering, and some of my classmates went into IT, some software engineering, some went into telecom, some went into traditional electrical engineering jobs (like power engineering, hardware design, etc.). I should also note here that if you have a passion for something, and can acquire some basic skills/knowledge (very easy to do in the days of the internet) in a field, your degree shouldn't be much of a barrier. For instance, I have degrees in computer engineering, computer science, and biomedical engineering, and I've worked in IT, hardware, and software research & development. You just need a good answer for the question "So, I see your background is in <X>, why do you want a job doing <Y>?"
Now, a bit about the traditional divisions of labor and different job titles...I don't think you'll find any strong consensus on these things - it really does vary from industry to industry and sometimes from company to company. For instance, sometimes the title "Engineer I" is a high position, sometimes its entry level. In some industries "Senior" Engineer is really the basic/most common position (especially commonplace in R&D where most new hires have an MS).
Here is my view of some of the types of jobs you listed:
IT is mainly support - fixing problems, interacting with users, building out infrastructure for servers, etc. Lots of jobs in lots of different geographic areas. Huge diversity in terms of pay, respect, and expectations.
Hardware spans a large range, but mainly relates to the design and development of computer architectures, ASICs, FPGAs, etc. These jobs seem (to me) to be fairly rare and clustered geographically. My undergrad specialty was "computer architecture", but I have never been able to even get an interview doing anything connected with designing computer architectures (mainly because there are so few companies that do that kind of work).
Telecom (or RF) Engineers: Typically deal with siting cell towers, building out network infrastructure, and managing the design and deployment of telecom networks. There seem to be a lot of jobs in this area, but localized to only a few companies (because there are only a few telecoms....)
Systems Engineer - This role varies, but it is typically hardware oriented and deals with all phases of development of large systems and "systems of systems" (like commercial airliners or submarines). Might interface or draw skills from industrial, mechanical, or software engineering. These are fairly common jobs, but also might be clustered geographically.
Software Engineer - Typically a software engineer deals with all phases in the lifecycle of a software system (requirements, architecture, implementation, testing, deployment, etc.). You might deal with really large software or software/hardware systems. Very common jobs, pretty much anywhere that there is significant numbers of tech jobs, a lot of them will be in software.
Software developer - This typically implies (to me anyway) a more "lightweight" role than software engineer - the focus here is on implementation, or less complex software systems (like web applications, etc.). Same job situation as the SWE.
Regarding management positions - in my industry (software R&D) everyone is technical, all the way up to the level of Vice President of Research. I'm sure there are a few people that have MBAs, but everyone also has an advanced technical degree (MS or more typically, a PhD), and management skills are taught (and there is not usually a shortage of highly motivated technical people who can also "pick up" the skills required for management).
Edit: Lots of typos...