The Connector's are "shared" by all Hosts and all applications. So the connectionTimeout is already global for all requests that come in on the corresponding Connector port.
This parameter is there specifically to fight one type of Denial-Of-Service attack, whereby some malicious client(s) create a TCP connection to the server (which has the effect of reserving some resources on the server for handling this connection), and then just sit there without sending any HTTP request on that connection. By making this delay shorter, you shorten the time during which the server resources are
allocated, to serve a request that will never come.
This has to be balanced against legitimate clients which may be slow in sending the request line. But 60 seconds is a really long time to wait for such legitimate client requests nowadays. You can probably lower that to 5000 (= 5 s.) without any ill effect on the legitimate clients.
Note that as soon as the HTTP request line has been received, this timeout plays no role anymore. So it is not usable to limit the time that an application requires to process and respond to the request. As far as I know, there is no standard parameter available in Tomcat to do that. Which is also rather understandable, because Tomcat has no idea what kind of delay makes sense for any particular application with any particular request parameters. Only you would know that, on a call-by-call base.
Final note : if a browser makes a request to a HTTP server, and does not get any response for about 5 minutes, the browser will time out and show an error message like "the server is not responding" etc..