you can say but it's more casual. You can say: I have a bit idea or I have a little bit idea

It's in common use where I come from in England. Casual and concise :)

I’ve heard “have a little thought” but never “have a little think”

In normal conversation you would just say "I thought about" or "I've been thinking about". But I (native speaker, American) do sometimes say "Have a think about that" or "Thanks, I'll have a think about that"--it's kind of a cute (whimsical) way to say the same thing. It's not particularly common (perhaps more common in the UK?) but you'll certainly be understood. But as others say, the default way to say it is just "think about" or "consider"

That sounds quite natural to me, if perhaps a little old-fashioned! It would probably be more common in daily conversation to say something like, 'I've been thinking a bit about that' or 'I've been thinking a bit about taking a trip to Europe.' Or even, 'I've been considering taking a trip to Europe.' Another (slightly different) expression you could also use: 'I've been thinking it over'/'I've been thinking over a trip to Europe' -- that would mean something like you've been considering the pros and cons of doing something.