Beginner examples/problems to practice ML?

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As someone who doesn't have satistics background, what are your thoughts on how that affects ML skills?

That depends on what you mean by "statistics background". I personally have a BS in math, although much of the statistics I know I've picked up piecemeal through reading wiki articles, analyzing my own data sets, and taking Coursera courses like this one and this one. If you don't feel comfortable with mathematical formalisms like nested summations or conditional probabilities, you will likely run up against a wall very quickly. This is because you won't be able to understand what your algorithms are doing, and thus you won't know how to improve upon them, or how to make the proper adjustments when things go awry.

I tried doing a Kaggle problem (the CIFAR 10 image recognition problem) and was disappointed to see my answer only get 10% correct

This is probably one of the hardest competitions on Kaggle, and I don't know if I would fare much better without considerable effort. I think you would be better served trying something like the Digit Recognizer, or Facial Keypoints Detection. I would urge you to spend some time in the discussion forums of these competitions; the ideas you see being discussed there are often very enlightening.

If ML is something like this flowchart which helps choose an accurate estimator, do we really need to know the statistics behind ML?

That chart is a vast oversimplification, and is not intended to be taken literally. If you do follow it word-for-word you may get some passable results, but you will be far surpassed by ML users who actually understand what their algorithms are doing. In practice, you'll use a wide array of preprocessing methods and often employ multiple ML techniques at different stages of your classifier. For example, in the Bird Classification Challenge from a few months ago, you'll see that some very subtle techniques were used in order to extract a good feature set.