How do I use existing cat5e wiring to distribute internet in my house?
Ho Le Crap! most of the pics you are showing involved phone service, not in house Ethernet / LAN. Your 6 pin connectors are for RJ16 jacks/plugs (3 line phone service).  RJ45 requires 8 conductors and an 8 pin jack/plug. cat5e or cat6 can be terminated on a patch panel, but not a punch down block ("A" in your pics). 

It can be pretty simple:  Connect all the cat5e cables to a patch panel and then choose which ones go to a router/switch.  Alternatively, you could simply  not use a patch panel and just install plugs on the ends of the cat5e cables and plug them into a router or switch.   

Bottom line:  You need to separate your Internet (coming over POTS - Plain Old Telephone Service) from your Ethernet connections.