You're the same height as me. I know it's a bit short, but 63 kg is a healthy weight at that height.
Now, what you want is some more body weight, but with a good distribution. Unfortunately, you can't influence the distribution of your body fat. Where it goes to is genetically determined. For men, fat tends to accumulate more on the lower belly. This is the increase in your tummy you're seeing. It gains fat first, and is also the last to lose it. For your body, this is your "emergency backup" it keeps around in case you're starving.
What we can influence is the distribution of body weight for muscle. Gaining muscle is a lot harder than gaining fat, but it also is distributed much more nicely and will make you look a lot better. Since you're willing to eat enough, you could do very well on a beginner strength program. Look into the term "linear progression", learn about the compound movements (squat, deadlift, bench press, overhead press, chin-ups, pull-ups, rows...) and check out some programs such as Starting Strength and StrongLifts 5x5, which have the right idea for a beginner strength program.
The weight you'd gain on strength training would be a lot more useful. There would be some fat, but also muscle mass. The fat can be reduced later.
As for the shape of your face, there's not that much you can do about it. I don't like to tell people "be happy with who you are" because I believe it's a good thing to want to improve. But don't spend time being unhappy or anxious over things you can't change. If you can gain some muscle and look less skinny, more "filled out", you'll look better overall. People will notice, trust me. Also, I'm not sure if it's true, and its only anecdotal, but I've had people tell me my face shape looks better since I've started strength training.