This looks like a firewall problem. Port 443 is very common (used for SSL and HTTPS) and is allowed through almost all firewalls but port 3333 is not. Keep in mind that most run-of-the-mill / low-end routers and firewalls will pass almost all ports and therefore work fine for mining. If you are running a higher end firewall, then you typically must configure it manually to allow ports like 3333 and 3334 to work.
A standard home firewall blocks incoming traffic. Outgoing connections would not be affected. If you had a high end router that could do more, you would know how to use it already and have discounted this as a possibility.
You arent getting a response when pinging with "ping -n 50 -l 128 speedtest.usa.nicehash.com:3334". Of course you arent getting a response, you dont ping ports, you ping IPs.
As Fanatic points out, you cannot ping to a port but you can ping the address or the hostname i.e. "ping stratum.antpool.com". That will tell you that the host is alive and reachable from your network. However, this test will not test the connectivity to port 3333 or any other port for that matter. You can test connectivity to specific ports using the "telnet" command.
Example "telnet stratum.antpool.com 3333 <enter>". If you do this, either one of two things happens - you get an error message of some sort (a windows machine will give you a "connection failed" message), or you will get a connection (generally indicated by the screen clearing and returning to a command prompt after you type a character or two). If you get the connection failed message, you are either trying to connect to a non-existing system, have the wrong port, or you are being blocked by a firewall. If you get a connection using telnet, then you should be able to connect with your miner.
