With regards to the problem with ICMP from the DMZ/LAN to the external network the problem is most likely the missing NAT configurations. You should have a Dynamic PAT configurations for those internal subnets so they are visible with the public IP address to the external router. Then the ICMP should work
You could for example use
nat (any,outside) after-auto source dynamic any interface
I am not sure about the problem with the LAN -> DMZ ICMP problem. If this is a simulated lab situation I would make sure that the R1 has a default route towards the DMZ interface IP address of the ASA.
There does not seem to be anything in the ASA configurations that should prevent ICMP from LAN to DMZ.
Table of ContentsIntroductionVersion HistoryPossible Future
UpdatesDocuments PurposeNAT Operation in ASA 8.3+ SectionsRule Types
Network Object NATTwice NAT / Manual NATRule Types used per SectionNAT
Types used with Twice NAT / Manual NAT and Network Obje...
[toc:faq]Introduction:This document describes details on how NAT-T
works.Background:ESP encrypts all critical information, encapsulating
the entire inner TCP/UDP datagram within an ESP header. ESP is an IP
protocol in the same sense that TCP and UDP are I...