You are a bit confused about the nature of addresses in 802.11 frames. Since your question isn't actually about the address fields, let me address that first and then go into the details on the address fields later in my answer if you are interested.
Most devices to which you are referring (whether they are call boosters, repeaters, extenders, etc) don't actually play a part in 802.11 as either a station or part of the distribution system (DS, also most commonly the access point or AP). They are simply devices that listen for and then repeat a signal.
For example, let's say Bob and Bill need to pass a message by radio. Unfortunately, they are too far apart (or maybe there is a mountain range in between) so they cannot reach each other directly. Ben happens to be in the middle (or on the top of the mountain) and since Bob and Bill can't directly talk to each other, Ben offers to repeat their messages so they can communicate.
So when Bob says "Hey, Bill, how are you?" on his radio, Ben hears it and then repeats it back into the air and Bill then gets the message. Any replies from Bill are handled the same way. Ben is not the source, the destination, or the intended receiver of the message. He doesn't really directly play a part in the conversation at all. This is the same process a "booster" will typically use on a 802.11 network.
The way the address fields are used in an 802.11 header is determined by the status of the "To DS" and "From DS" flags.
Address 1 will always be the receiver address (RA) and address 2 will always be the transmitter address (TA). This provides a consistent way for radios to easily determine if they need to receive the frame.
If neither of the "To DS" or "From DS" flags are set, this is a management or "station to station" frame. In these cases, address 1 is is the destination address (DA) in addition to the RA. Address 2 is the source address (SA) in addition to the TA. Address 3 would contain the BSSID (so radios not in the BSS can filter it) or be set to broadcast (all F's). Address 4 is unused.
If only the "To DS" flag is set, this is a frame bound from a station to a DS. In this case, address 1 is both the RA and the BSSID, address 2 will be both the TA and the SA, and address 3 will be the DA. This makes sense as while the station needs to send the frame to the DS, it is actually destined for some other host on the network and not for the DS.
If only the "From DS" flag is set, this would be a frame from the DS (or AP) to a station. Address 1 will be both the RA and the DA, address 2 will be both the TA and BSSID, and address 3 will be the SA. Again this makes sense as while the DS is transmitting the frame, it was sourced from another device on the network.
If both the "To DS" and "From DS" flags are set, this would be frames sent from one DS to another DS, such as you would find when using wireless access points to bridge traffic for other devices. Address 1 will only be the RA, address 2 will only be the TA, address 3 will only be the DA and address 4 will only be the SA.
Good question. I'll answer it with an animation:
When Host A sends the frame, the switch does not have anything in its MAC address table. Upon receiving the frame, it records Host A's MAC Address to Switch Port mapping. Since it doesn't know where the destination MAC address is, it floods the frame out all ports.
This assures that if host B exists (which at this point, the switch does not know yet), that it will receive it. Hopefully, upon receiving the frame, Host B will generate a response frame, which will allow the Switch to learn the MAC address mapping from the return frame.
You can read more about how a Switch works here (where I took the animation from). I would also suggest reading the entire article series for a closer look at how a packet moves through a network.
One last note regarding the terms Flooding vs Broadcast. A switch never broadcasts frames, a broadcast is not an action a switch can take. A switch can only flood a frame. A broadcast is simply a frame with a destination MAC address of ffff.ffff.ffff. This is often confused because the end effect is the same, but they are actually different.
Best Answer
MAC addresses are layer-2 addresses in the frame header. The layer-2 frame encapsulates the layer-3 IP packet. The layer-2 frame is stripped off at a layer-2/3 boundary (e.g. a router). If the layer-3 packet needs to be forwarded through another layer-2 domain, a new layer-2 frame is created using MAC addresses in the new layer-2 domain to encapsulate the layer-3 packet.
Conceptually:
To the router:
<L2 frame><L3 packet><L4 segment>PAYLOAD DATA</L4 segment></L3 packet></L2 frame>In the router:
<L3 packet><L4 segment>PAYLOAD DATA</L4 segment></L3 packet>From the router:
<New L2 frame><L3 packet><L4 segment>PAYLOAD DATA</L4 segment></L3 packet></New L2 frame>The original layer-2 frame will contain the source MAC address of the sender, and the destination MAC address of the router's interface in that layer-2 domain.
The new layer-2 frame will contain the source MAC address of the router's interface in the new layer-2 domain, and the destination MAC address of the next hop in the new layer-2 domain.
The layer-3 packet will contain the layer-3 source IP address of the originator of the packet, and the layer-3 destination address of the final layer-3 destination.
Layer-2 MAC addresses are removed at each layer2/3 boundary along the routes to be replaced with the MAC addresses of the new source and destination, but the layer-3 IP addresses stay the same along the path (with certain exceptions like NAT).