Do the stock performances of various companies have a measurable correlation with exchange rate? For example if the values of big companies in a country decrease, will the value of that country's currency measurably decrease?
A company valuation is merely in paper. It is similar to  economist eating shit jokes. When the stock price of a big company drops, it "suppose to" hurts the shareholder who bought it on high.  

When I say "suppose", because intervention does happens to "too big to fail" company. In worst case, government hold too much "interest" in big company that might issue bonds(AKA print money) to bails out the entity, as what happens to Asian financial crisis, and USA subprime loan financial crash. 



IMHO, details real world example falls under TL;DR; category. There is countless global example. 

A large company stock can be jack up by speculation. Say a company issues  10 millions stock with $1 face values. If a willing buyer trade 100 stocks with $1.50, and subsequent buyer keep buying 100 stocks and jack it to $50, it doesn't mean the company "values" sudden shoot up from $1 x 10 millions = $10 millions  to $50 x 10 millions($500 millions). This WILL NOT create extra $490 millions to the economy, nor to the money supplies.  

Without government money supplies and cheap loan rates, it is impossible to create such speculation. In addition, without increase money supplies, not hot money flow into a country, foreign exchange will always stay in a stabil level.

Example of high stock values and insane inflation 


Zimbabwe stock market compare stock value.


Examples of bubbles crash and NO bailout made, no apparent inflation 


Dot Com bubbles.
Enron, Worldcomm scandal. 


Example of bubbles crash. Country bail out cause higher inflation 


Asia pre 2007 economy crisis stock value


Example of crash and bail out inflation flow to other country. 


USA QE. Speculator borrow cheap money and speculate outside US. Hot money cause many Asian country company stock shoot up without fundamental.