By your description you are on iOS 14 beta as that's the first release with widgets and an app drawer.
Back swipes are app dependant, if that developer didn't want them you don't get them.
Try a different keyboard app, several like touchpal allow increased keyboard size.
Widgets are half ass because it's a beta and developers need to make widgets for their apps.
If you are in the message thread that picture and name let's you know week you are messaging, this is like any messaging app, you can turn off contact photos in settings under messaging. If you are in the message list is because you have pinned contacts that you are most active on. Clicking on the picture sends you to the message thread. You can unpin contacts by long pressing on the in the message list.
I search by adding Siri so I haven't ran into the search issue but I can see how that could be counter intuitive.
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PSN: Diemenz
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