Hi! Let's just say different versions of Docsis supports different speeds. You don't have to know what exactly it means, but the the newest standard on the market is 3.1. However, I'm pretty sure her speeds are more on the slow side since you said she likes saving money. You should go with a Docsis 3.0 one (her current one is also docsis 3.0)

A splitter looks like this https://www.amazon.com/Valley-2-Way-Cable-Splitter-Video/dp/B001X3MGFU and is used to split a signal so you can connect one line to both tv and internet modem. These are used a lot, but since its splitting a signal (hence the name) you are WORSENING the signal to the modem which can eventually lead to more issues.

I did a quick search on amazon, and most Docsis 3.0 modems WITHOUT voice is around 50 or even under. However, immediately after putting VOICE MODEM after it the price jumped to over 150, some going over 200.

That modem seems like its from comcast. They might give her a new one but she'll have to pay $14 a month for modem rental.