Ip – Network Device sending ARP request (opcode 1) but with Source IP 0.0.0.0

You're almost 100% correct. Slight corrections bolded below:

1. HostE will send an ARP Request with a target address of Router2's IP address. The destination MAC address of the ARP packet will be ffff.ffff.ffff, which would indeed make it a broadcast. Router2 responds unicst with its MAC address, and HostE generates the packet with a Source and Destination IP of 192.168.3.1 and 192.168.1.1, and a Source and Destination MAC of 7777.7777.7777 and 8888.8888.8888 (respectively).
2. Router 2 will receive the packet, and consult its Route Table to determine the Next Hop for the 192.168.1.0/24. If Router2 doesn't already know Router1's MAC address, it will issue a broadcast ARP Request for it. Once it learns it, it will then forward the packet leaving the Source/Destination IP unchanged, and encapsulating the IP packet with a Layer 2 header with a Source and Destination MAC of 5555.5555.5555 and 3333.3333.3333.
3. Router1 receives the packet and consults its Route Table to determine the destination Network is directly connected. Router1 sends a Broadcast ARP Request for the IP 192.168.1.1 to learn HostA's MAC address, and then finally forward the packet to the final destination.

And to answer your last question. Whether HostE gets its address from DHCP or not, the process flow would appear as above if HostE did not have an entry in its ARP Cache for Router2's MAC address.

IF, Router2 was the DHCP Server (which it doesn't have to be). And Router2 just assigned HostE its address. Then theoretically, HostE would know Router2's MAC address. But ARP entries don't last forward, particularly on clients. For example, my Windows 7 laptop keeps ARP entries for only 34.5 seconds (see below).

So even if Router2 was the DHCP server, and gave HostE its IP address, sooner or later HostE's ARP cache would time out, and it would have to run the broadcast ARP request all over again, just like if HostE had always only had a static IP assigned.

C:\Users\eddie>netsh interface ipv4 show interfaces 20

Interface Local Area Connection Parameters
----------------------------------------------
IfLuid                             : ethernet_11
IfIndex                            : 20
State                              : connected
Metric                             : 20
Link MTU                           : 1500 bytes
Reachable Time                     : 34500 ms     <-----
Base Reachable Time                : 30000 ms
Retransmission Interval            : 1000 ms
DAD Transmits                      : 3
Site Prefix Length                 : 64
Site Id                            : 1
Forwarding                         : disabled
Advertising                        : disabled
Neighbor Discovery                 : enabled
Neighbor Unreachability Detection  : enabled
Router Discovery                   : dhcp
Managed Address Configuration      : enabled
Other Stateful Configuration       : enabled
Weak Host Sends                    : disabled
Weak Host Receives                 : disabled
Use Automatic Metric               : enabled
Ignore Default Routes              : disabled
Advertised Router Lifetime         : 1800 seconds
Advertise Default Route            : disabled
Current Hop Limit                  : 0
Force ARPND Wake up patterns       : disabled
Directed MAC Wake up patterns      : disabled

The switch, while a layer-3 switch, is only a layer-2 switch for a LAN. As a layer-2 switch, the switch doesn't get involved with the ARP request/reply, unless it is the target of the ARP request, or it is making the ARP request. Of note is that the SVI address of the layer-3 switch is just another host on the LAN, and it will update its layer-3 ARP cache when it sees the ARP request, but it will not actually interpose itself in the process between the two hosts.

The switch will also update its MAC address table for the LAN with any frame coming into any of its layer-2 interfaces.