Looking at about:config
on Firefox 33 on GNU/Linux (Ubuntu), and searching connections
I found:
network.http.max-connections: 256
That is likely to answer the part is there any limit to the number of active connections per browser, across all domain
network.http.max-persistent-connections-per-proxy: 32
network.http.max-persistent-connections-per-server: 6
skipped two properties...
network.websocket.max-connections: 200
(interesting, seems like they are not limited per server but have a default value lower than global http connections)