I have my lightroom on an external hdd connected via USB 3, and I switch between my desktop and laptop with some regularity. My question though is regarding Lr Mobile. Will it sync normally from both computers with my iPhone? In other words, if I sync it on my desktop, and then a few hours later edit photos on my phone, and then open my library on my laptop, will it cause problems?
Lightroom Mobile doesn't sync all photos, it only sync's with photos in Collections you have configured to sync. If you change an image in a given collection, it should sync, regardless of where you make that change, or even what Lightroom database you are using.
You need to distinguish between your 'photos' and the Lightroom Catalog. Since Collections are defined within the Lightroom catalog, your images on the hard drive are not really what is being sync'd, but rather the metadata. So, if you use the same catalog between your laptop and desktop, then it does not matter which computer made the edits, they will appear from the mobile app when you open the catalog. However, if you have separate catalogs per machine, then the edits to the images in the Collection will only be associated with the catalog that created them.
So,ensure the catalog used by each machine is the same and it should work fine. (for example, keep catalog on the external hard drive.)