Statistics of links between public debt and recessions

Yes actually many people bothered to study this link. To be more specific the studies are between debt-to-GDP and economic growth (but recession is defined as negative economic growth - more narrowly by NBER as two consecutive quarters of negative growth). Here is a brief summary of the recent literature:

Rogoff-Reinhart (discredited) work:

As mentioned in the title the work of Rogoff-Reinhart is now discredited due to several embarrassing mistakes they made in their analysis but any contemporary mention of literature review on this issue would be incomplete without mentioning them as they arguably brought attention to this topic.

Rogoff-Reinhart research is non-technically summarized in their book This Time Is Different: Eight Centuries of Financial Folly. They did quite a comprehensive study of a historical records and also very meticulous collection of historical data on government debt statistics. Analyzing this data they argued that there was a 'magic number' of debt to GDP ratio of $90%$. They argued that below this cut-off debt did not had much effect on GDP growth but above it it started to have a negative effect.

However, it was later during replication discovered that Rogoff and Reinhart made several 'embarrassing' but probably unintentional mistakes in their analysis and replication of their study found that their result virtually disappeared when these mistakes were corrected. If you are interested in knowing what the mistakes were you can read this Economics.SE answer).

Post Rogoff-Reinhart Research

However, while Rogoff-Reinhart work was discredited it brought much needed attention to this issue. Further studies that paid attention to cross country heterogeneity, endogeneity and avoided arbitrary debt-to-GDP brackets suggest that the correlation between debt-to-GDP and growth is actually still negative but the issue is also much much more nuanced than suggested by original Rogoff-Reinhart work with their 'magical' threshold. It is also still up to debate what exactly is the nature of causality here and it is agreed that even if there is a causal link the relationship is definitely non-monotonous.

A nice survey of post Rogoff-Reinhart literature can be found in Panizza and Presbitero 2013. Some important papers that you should read if you are interested in this include Minea and Parent (2012), Egert (2015), Panizza and Presbitero 2014 or Baglan and Yoldas (2013). There is also this VOX$^{EU}$CEPR article which makes also short summary of this area. You can find more papers in sources cited therein or by searching for keywords from those papers.

When it comes to civil unrest I would recommend Politics.SE or some sociology forum as such issues are not normally studied by economic profession or if then in a interdisciplinary way.