Hi, Crinklynose, I started out renting one for my daughter for a couple of months. She broke one of the pegs, the rental shop sent it in for repair and didn't have her size for her.  At that point I thought she would stay with it and like Poke said, it was cheaper over the course of one or two years so I bought a Cecilio cvn 500 for her. I personally never like to rent anything, because you pay the $ and at the end you don't get any thing. Like poke said, your rental probably sound better than Cecilio.

When my daughter started to play violin, most people suggested us to rent one for her because "children grow out of their violins fast." After two years, I have learned that children stay with their violin (if you child grow about four inches a year) for one to two years or longer(if your child grow about two inches a year), depends on how fast your child grow and how long their arms are. Say if your child used a violin for two years and you pay $20 rent per month, you end up paying $480. With that price you could get a pretty decent violin and in the end you could sell that violin or trade in that violin (the better violin you buy the higher the resell/trade in value).

For your daughter, I don't know if she is growing out her 1/8 yet, if she is growing out of it soon, and if you still prefer to buy one for her I would say buy a 1/4 size when she needs a bigger size. Like poke suggested, you could have the FM pick one for you.

The bigger the violin, the better/bigger it would sound, not that it would sound like a full size violin, but it would sound bigger. As she improves, she would play better and it would sound better as well.

Yes,  the distance between strings for 1/8 is much closer than distance on bigger violin. However, I kept touching wrong strings still - my muscle memory is not that good. I would say as she practices, she will figure out how to control her arm so her bow would not touch the wrong string.