Can I use http tunnel to ping or traceroute through a proxy with firewall?

It's not exactly what you were looking for, but if you can access and external ssh server, you can run it through that, and the results will reflect the ping time to the ssh server:

$ ssh username@server 'ping -c 1 google.com' 
PING google.com (72.14.204.147) 56(84) bytes of data.
64 bytes from iad04s01-in-f147.1e100.net (72.14.204.147): icmp_seq=1 ttl=57 time=2.64 ms

--- google.com ping statistics ---
1 packets transmitted, 1 received, 0% packet loss, time 0ms
rtt min/avg/max/mdev = 2.640/2.640/2.640/0.000 ms 

No. Ping and traceroute make use of lower layer network protocols (ICMP and/or UDP, in particular, which are layer 4 protocols) and will not work over an HTTP (layer 7) tunnel. In any case, even if you could convince the HTTP proxy to open a raw TCP session for you (which is how some HTTP tunneling works) you would not receive the necessary packets to confirm that the host is reachable. (ICMP echo reply, in the case of ping, or the time-to-live expired ICMP packets in the case of traceroute)

To test for connectivity in this situation, I think the best you can do is an HTTP "ping". (Try to establish an HTTP connection with the remote host and see if it works.) For example, you could do something like:

$ http_proxy=http://webproxy.example.com:3128 \
> curl -I http://google.com/ > /dev/null 2>&1 \
> && echo success || echo failure

Assuming you have curl installed, this would print "success" if google.com is reachable through your proxy and "failure" if not.