Are there any types of meditation/mindfulness intervention not starting with breathing exercises? It occurs to me that all meditation/mindfulness practice starts with focusing attention on breathing by either counting breaths or simply feel the sensation of the airflow 
(Baer, 2003). Eventually, this focusing attention on breathing is likely to induce deep, slow breathing, which can slow down the heart rate and calm down the nervous system. Is there any meditation practice starting with a different form other than focusing on breathing?

Baer, R. A. (2003). Mindfulness training as a clinical intervention: A conceptual and empirical review. Clinical psychology: Science and practice, 10(2), 125-143.
You can use key words (“body scan” combined with “meditation”) to get specific studies. Dr. Jon Kabat-Zinn’s book (titled "Full catastrophe living: Using the wisdom of your body and mind to face stress, pain, and illness”) describes this body-scan meditation and also you should be able to get youtube videos on it. Although it is typically described for lying down position, it can be done sitting or even standing. 
I found the following study by doing a quick web search – there could be newer studies:

Ditto, B., Eclache, M., &amp; Goldman, N. (2006). Short-term autonomic and cardiovascular effects of mindfulness body scan meditation. Annals of Behavioral Medicine, 32(3), 227-234.

Walking meditation is also very good if one tends to fall asleep when meditating. Here’s a study that has used walking meditation:

Prakhinkit, S., Suppapitiporn, S., Tanaka, H., &amp; Suksom, D. (2014). Effects of Buddhism walking meditation on depression, functional fitness, and endothelium-dependent vasodilation in depressed elderly. The Journal of Alternative and Complementary Medicine, 20(5), 411-416.

Here’s ‘soles of the feet’ meditation – again, there could be newer references:

Singh, N. N., Wahler, R. G., Adkins, A. D., Myers, R. E., &amp; Mindfulness Research Group. (2003). Soles of the feet: A mindfulness-based self-control intervention for aggression by an individual with mild mental retardation and mental illness. Research in Developmental Disabilities, 24(3), 158-169.

Another meditation technique is to be simply aware of sounds, etc., that come and go without getting entangled in thoughts – this is called choiceless awareness and this meditation is also described in the book of Kabat-Zinn I mentioned earlier. 
Here’s a reference for that as well:

Brewer, J. A., Worhunsky, P. D., Gray, J. R., Tang, Y. Y., Weber, J., &amp; Kober, H. (2011). Meditation experience is associated with differences in default mode network activity and connectivity. Proceedings of the National Academy of Sciences, 108(50), 20254-20259.