Lets call the switch between subnets 2 and 3 S2. That is, router R1 between subnets 2 and 3 is now replaced with switch S2.  
 
a) No. E can check the subnet prefix of Host F’s IP address, and then learn that F is on the same LAN segment. Thus, E will not send the packet to S2. Ethernet frame from E to F: Source IP = E’s IP address Destination IP = F’s IP address Source MAC = E’s MAC address Destination MAC = F’s MAC address  
 
b) Yes, because E would like to find B’s MAC address. In this case, E will send an ARP query packet with destination MAC address being the broadcast address. This query packet will be re-broadcast by switch 1, and eventually received by Host B. Ethernet frame from E to S2: Source IP = E’s IP address Destination IP = B’s IP address Source MAC = E’s MAC address Destination MAC = broadcast MAC address: FF-FF-FF-FF-FF-FF. 
 
c) Switch S1 will broadcast the Ethernet frame via both its interfaces as the received ARP frame’s destination address is a broadcast address. And it learns that A resides on Subnet 1 which is connected to S1 at the interface connecting to Subnet 1. And, S1 will update its forwarding table to include an entry for Host A. 
 
Yes, router S2 also receives this ARP request message, and S2 will broadcast this query packet to all its interfaces.   B won’t send ARP query message asking for A’s MAC address, as this address can be obtained from A’s query message.  
 
Once switch S1 receives B’s response message, it will add an entry for host B in its forwarding table, and then drop the received frame as destination host A is on the same interface as host B (i.e., A and B are on the same LAN segment).