Philosophy, love it here we go...
There is a place named Auroville where money does not exist, yet people are still happy to live there. But you probably aren't after an answer which is unrelated to normal 'western ideals'.
You can read an entire article written by Australian Unity, the makers behind the Australian Unity Wellbeing Index, on this question.
But here is my take:
While you do need money to get your basics going, it doesn't go much further than that:- A roof above your head, some food, essential items, etc. These material things really don't increase happiness, but are somewhat necessary. Its sort of the foundation - its needed, it sustains everything on top of it, but no matter how fancy it looks, its still a foundation which you cant see. It makes no difference.
Having money (or fame) attracts girls, yes. But, not the kind that you really want. Run out of the money (or fame) and they all go running away - let them in too much and they'll stab you in the back. Sure, that isn't representative of everyone, but it does happen much more than expected.
If your after a meaningful relationship, you wont need money. If your after some fun for a day and move on, just go to a bar.
What makes you happy are the things you spend most your life on. Your job, and your relationships (if any of those apply). If you don't like your job, you probably wont be happy. If you love your job, you'll be happy.
One last thing on online personalities. No-one who wants online attention will show you what terrible things there are in their life. They will usually only show the best parts, and the most enjoyable. The same goes with married couples - yours isn't going too well in your eyes, but all your friends have such good relationships. 4/5 times, they will be thinking the same thing with their relationships. It is all a matter of perspective.
Conclusion, it can buy happiness. But, it has a small limit and there are other aspects such as work and relationships (With other humans, or pets) which influence your happiness a lot more.