In the macro settings of a lens, f5.6 gives you a razor thin DOF only, which means you can "fall out of focus" by only just breathing, or having a heart beat (which are two things that you cannot avoid, ok, you can hold your breath, but hold your heart beat?????

Not quite). So yes, I think overread is right in saying that a smaller aperture is better for the macro settings. You will STILL get a blurred background, it's not like DOF becomes metres deep in the macro settings!
However, with that smaller aperture, you automatically get a slower shutter speed. In case you have to work with a fully extended zoom lens in order to get to that lens's macro settings (like I have to do), slow shutters can easily, very easily cause camera shake. More so, if your flower in question is in the shadow.
While I, up to now, have never ever so far taken any of my macros with the camera on the tripod, it
is recommendable, I cannot but agree! (I find it to cumbersome to haul my tripod into our flowerbeds, in amongst bushes and twigs and all, but that doesn't mean it would not give me better results if I did!)
Much of it is practise and more practise. You can also learn to get a steady hand, to really hold your breath for the time you need to focus. Today, you don't have the bin full of blooper prints any more in order to practise and practise more. Was different with film and prints ... boy, I cannot count the bloopers I shot and had to pay for just so I could throw them away...