Is it possible to write morally grey characters with happy and bright personality, and not always gloomy and dark all the time like in most common fiction?

Oh, absolutely.

I have plenty of morally grey characters in my books. Some of them are gloomy and dark (Felix Apollytes comes to mind; Jacques Dumond has also been known to do some world-class brooding), but many of them are quite cheerful.

For example, there's Katrina Stormwind. Mostly we see her when she's causing havoc, but under normal circumstances she's quite a happy young woman.

Or you have the best-kept secret in Hell, an assassin who's in love with a demon. He's pretty content with his life.

Cian, the hero of my most recent novella, is definitely morally grey. Google had…concerns…when I was researching something in conjunction with Cian beating a guy to death. Cian is also a pretty happy-go-lucky kind of guy who just happens to have a fascination with knives. Most of the scenes he shows up in before this latest book depict him flirting with his on-again-off-again ex-girlfriend, who broke up with him in the first place because she didn't approve of his extracurricular activities—and she's an assassin.

And let's not forget Kakra and Zesi, twin bundles of mischief whose driving motivation is to get laid, much to the dismay of their brother-in-law who's tasked with looking after them. They're also equally happy to kill people who make the mistake of threatening them.

I'm not sure I'd call Felix’s love interest morally grey, although she does like to joke about eating babies. Plus she helped torture someone at one point, but he deserved it, so there's that.

But anyway, yes, it's absolutely possible to write a morally grey character who isn't always battling their inner demons but rather embraces themself for who they are.