Re: Wading in late
A few points:
When Hartnell morphed into troughton they hadn't even come up with the word "regeneration" - they were originally going to go with "renewal". They had started thinking about replacing Hartnell a while earlier, as his health was failing and he was having trouble getting his lines right (in those days it mattered more, as they couldn't spend time doing a new take unless the first go was completely messed up). It's the need to produce the show quickly and with low cost that gives the earlier series an unpolished feel.
The educational focus of the show was more on history, not science. Pure historical episodes were a feature of the first few seasons. Later on they realised that just dropping the doctor and friends into the past didn't really generate much interest, so the pure historicals were dropped in favour of "Doctor goes back into earth's past and finds that <historical_event> was actually caused by ALIENS!". They still regularly fall back on this latter idea now (Shakespeare with alien witches, Robin hood with alien robots, and many, many cases of byegone Zygons).
The show was originally recorded on tape, it wasn't just produced for immediate consumption and ditched straight after transmission. Unfortunately, the magnetic tape of the time was horrifically bulky and expensive, so archiving rather than reusing tape was very costly. Episodes of the early series were lost because they needed to reuse the tapes for something else, that's all. A lot of the surviving episodes of the older series were actually sourced from copies that had been transferred to film so they could be shown by overseas TV stations. In many cases the soundtracks for these episodes were recovered separately; apparently some fans used to record the soundtracks off the air onto reel to reel tape so all the soundtracks still survive. If you really feel the need you can even watch the missing episodes in "slideshow" format alongside the recovered audio - they had someone who took a series of photos through each and every episode and these photos still exist. There are people who are still looking through old cupboards in foreign TV stations looking for film copies of the episodes where the visuals are still lost - amazingly they still make (re)discoveries from time to time. The amount of effort that's been expended to reconstruct the earlier series is mind boggling, and a testament to the show's cult appeal.
The original series wasn't about the Doctor and his sidekicks, instead it was focused on Ian and Barbara, Science and History teachers (respectively) who happened to end up travelling with a mysterious doctor. The specific teaching credentials of these first companions was all about facilitating the show's educational credentials. They were no sidekicks - they were the stars of the show and most of the time were more important to proceedings than the Doctor himself. Hartnell's character gradually took more control over proceedings, particularly after the departure of the two teachers, but it's only really from Troughton onwards that the Doctor really became the main character driving the plots. Even then, it's still a bit churlish to call some of these characters sidekicks - while you wouldn't trust Bonnie Langford's character to change a lightbulb the Doctor has had his fair share of useful companions through the years.
You can blame the solitary season for Eccleston on the man himself - apparently so afraid of typecasting he agreed to one season only, refused to renew his contract, and has done his best to keep away from the show completely since (which is why you only saw him alongside Hurt, Tennant and Smith as a life size cardboard cutout). No great problem for me - I preferred Tennant anyway - but I do agree it was in part his big name appeal that gave the show the boost it needed to relaunch as it did.