What is more common for first timers: 1. going with a tourism company; or 2. going with a group; or 3. doing the walk on your own at your own pace?
Why does it matter what is "more common"?
It's a serious question.
The fact that one thing is more common than another is evidence that there are less common options. If we all liked and enjoyed the same things then there wouldn't be "more common" or "most common" there would just be "the".
Despite the protestations of research elsewhere, I don't think you're quite there yet. I will admit though, that "research" can be something of a dull word for some of us.
I prefer inspiration.
Therefore, I suggest that you read/watch/listen to as much content as you can and see if you can discover what speaks to you. Not what stages people walked, when and at what pace. What people saw, felt and experienced.
Suitably inspired, the decisions you are asking about making will become far easier.
If we were to go at our own pace, how difficult is it to get lodging at your next stop?
Two things here.
Lodging may be "difficult" but a certain amount of that "difficulty" depends on our own mentality. The idea of sleeping in a porch (very, very unlikely) may sound appalling sitting at home but after a tough day - imagine pouring rain, slipping and sliding, a lack of food, aching muscles - the prospect will seem so much better in reality. Things are relative.
(In reality, a bus or taxi to the next available accommodation is probably the most likely option)
Secondly, you mention "we". Speaking only from personal experience, the planning I do for a trip for "me" is very different from the planning I do for a trip for "us".
The "we" person or people should be involved in these decisions.
Oh. So many things will depend on time of year and location that such open, vague questions are impossible to answer.
Is just starting the walk with no pre-planned stops recommended?
I think such a recommendation would not usually be made to first-timers but that's not to say it isn't for you.
The fact is that the CF is such a well resourced route that it is entirely possible to be entirely spontaneous in relative safety and comfort. It may be wonderful. It may be miserable. And everything in between. You may not stay in the "best" places and you may inadvertently pass some "must see" locations. But it may be "enough".
Sometimes "enough" is the best.