How to make wireshark filter POST-requests only

Ping packets should use an ICMP type of 8 (echo) or 0 (echo reply), so you could use a capture filter of:

icmp

and a display filter of:

icmp.type == 8 || icmp.type == 0

For HTTP, you can use a capture filter of:

tcp port 80

or a display filter of:

tcp.port == 80

or:

http

Note that a filter of http is not equivalent to the other two, which will include handshake and termination packets.

If you want to measure the number of connections rather than the amount of data, you can limit the capture or display filters to one side of the communication. For example, to capture only packets sent to port 80, use:

dst tcp port 80 

Couple that with an http display filter, or use:

tcp.dstport == 80 && http

For more on capture filters, read "Filtering while capturing" from the Wireshark user guide, the capture filters page on the Wireshark wiki, or pcap-filter (7) man page. For display filters, try the display filters page on the Wireshark wiki. The "Filter Expression" dialog box can help you build display filters.

I believe you will have to put a capture filter for all HTTP traffic, and then put in a display filter for the http.response.code == 500
After you have found a response code, remove the display filter and then use the Follow TCP Stream -or- the Conversation Filter to find the related packets...