I am glad you attached a picture to your post. I was having a hard time visualizing what you're describing. Admittedly, I am having a hard time visualizing what you want too.
But none of what I'm stumbling around on really matters. Fortunately, I believe the answer to all of your questions is, 'yes':
Can we have a TKB styled component on a forum page, with tiles, where we don't have the Solution functionality and then have a secondary component which is the standard list where users can ask a question?
Yes - you can put a custom component on a forum page, with tiles, that looks like the TKB...but it would just be easier to pull in TKB content via the API into those tiles. The function of these tiles, I assume, is to promote certain articles on the forum page. And yes, right near, those tiles you can put an 'ask a question' widget too.
Or can we use a different forum component where we can add visual images to certain questions we raise so our members can see that they originated from us and it overrides the profile avatar in those instances?
You could do this also, but the custom component's flexibility for you to change visual images on the fly might be a bit tricky if you have to do it often and you do not know how to manipulate the code. And yes, you can override the profile avatar via some CSS sleight-of-hand
But all of this sounds like a lot of work....and for what? I understand that perhaps you may have an extremely customized and tailored digital customer experience, but is this more for branding / aesthetic purposes, rather than for increasing the success and health of your community? When I start visualizing this experience (with my somewhat feeble mind), I am just wondering what real shot-in-the-arm it will offer to your Community members that is so radically different from what maybe you currently offer.
Please dont take my comment as being a negative-nellie or naysayer. I just want to understand what you are trying to achieve from a business point of view.