LAN Splitter or a Switch?

would our Ting service (w/1000 Mbps) be challenged with the Express?

Actually, it is the other way around. The Express has only 100 Mbps Ethernet ports so the maximum speed that any device.....WiFi or Ethernet......connected to the Express will receive would be 100 Mbps.

The good news....or bad news.....depending on your opinion......is that the Arlo Base Station has only a 100 Mbps Ethernet port, so like the Express, it will only be able to handle up to 100 Mbps speeds.

For what it's worth, you could argue that the Arlo Base Station is or is not a router. It does control the connections to the cameras, so in that sense it does act like a router. Otherwise, you could say that the Arlo Base Station is simply a camera controller which connects to your network.....which already has a router.

In addition to Roger Wilmut1's excellent comments:

• Your AirPort Express will only support 100 Mbps connections, even with a 1000 Mbps Ethernet switch. However, it still would benefit you in the long run to get an 10/100/1000 Mbps switch.
• An Ethernet "splitter" or hub are older technology and are fairly rare today. You want to avoid these and use a switch. As Roger has mentioned you just need an "unmanaged" switch for this purpose.
• Regardless of whether you use a hub or switch, all traffic connected to this device has to go through the single Ethernet port on the Express ... and that will be your network circuit's bottleneck. Remember, any 1000 Mbps-capable device connected to the Express will be limited to 100 Mbps ... even the switch.

I don't think there is any such animal as a LAN splitter - that would imply sending the same signal to two different destinations (like an HDMI switch), which you can't do on a LAN. What you need is an unmanaged* ethernet switch such as this one:

Such switches have their own power supply: you simply plug the output from your router into one of the sockets (sometimes specified, sometimes it desn't matter) and leads to your other devices into the other sockets, and it looks after itself. It's completely seamless. Requests for, for example, web pages from any device plugged into it get the page sent back to the right device; you can have several devices connected and all doing different things without interfering with each other. (Of course heavy activity may slow things down if you reach the limit of your incoming connection.)

If you are placing a router. if that's what the Arlo is, into the LAN after another router such as the Airport it's important that you set it to pass DHCP (IP allocation) through - 'bridge mode'. If you have two routers both handing out IP numbers you are going to have problems.

*A managed switch would require you to allocate the IP numbers for each device connected to it, and you don't want to do that. An unmanaged switch does all that itself, passing through the IP numbers set by the router feeding it.

I have an Apple Express that is being used as an extender and need an additional LAN port for an Arlo base station.

You mentioned that you needed an additional LAN port for another Ethernet device.

IF.....you are using the AirPort Express to wirelessly extend the signal from another Apple AirPort router, you might not be aware that both the WAN "O" port and LAN <--> ports will operate as LAN <--> ports in this type of setup, so the Express will support two different Ethernet devices.

IF......you need to connect a third Ethernet device to the AirPort Express.....then.....you will need an Ethernet switch.

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Already answered that for you above.