Are there any parameterized problems in non-uniform FPT that are suspected (but not proven) to be in uniform-FPT?
In Downey and Fellows' 2013 book (Fundamentals of Parameterized Complexity; Section 2.2), they mention an example of a problem in non-uniform FPT (Graph Linking Number) and briefly discuss that it's open whether this problem is in uniform FPT. This problem seems fairly natural.

When considering non-uniform parameterized complexity, one can look at many different variants, and many of them do have some relations to (non-uniform) circuit complexity. As far as I know, this hasn't been explored much.

For an overview of some basic connections, you could have a look at a report that I wrote: An Overview of Non-Uniform Parameterized Complexity.