Is there a way in ArcMap Legend Properties to make the legend above look like the picture below?
In short, no - not without a lot of messing around (and even then you may only get close).
The legend tools don't allow you to remove the symbol from the legend (which is how the bottom example is laid out).
The easiest solution would be to do as @Tom said and convert it to graphics and make your adjustments from there.
If you wanted to keep it as a dynamic legend you'll need to set the symbols for your layers to be really small (around 1 point) and make them the same color as the background of your map. This will make them not show up in the legend. However, it will also make them invisible on your map - so you may need to add a copy of the layer you are symbolizing and only include that one in the legend. That way you can mess with the symbols and still keep the feature counts like you are looking for.
The rest is just a matter of setting the shelters layer to display in 2 columns and placing the County Boundary underneath Shelters so it shows up on the bottom.
My apologies if that is confusing - messing with ESRI legends is not always straightforward.
Unless you're making a quick map you should create your legend manually and not with the insert legend process. By using custom text and graphics you can completely control your legend. The only downside is your legend won't automatically update when you update symbology. I make over 300 maps a year and only insert a legend if it's a rush job.