Yes, Elgg is perfect for that. My employer belongs to a group of 6 companies located in 4 different cities. Elgg allows us all to easily share information (posts, images, videos, etc) with each other and to interact with it (likes, comments, polls, etc).

For a small sized network, any shared hosting will do. I would go with Arvixe since they offer Elgg installations.

While I lack the experience that both Juho and Cim have with Elgg, from what I've seen and my limited experience with Elgg I would say it would perfectly for you.

For hosting I use GoDaddy and Elgg runs fine with their shared Linux hosting. GoDaddy doesn't offer Elgg installations so you have to do it manually which isn't a huge deal if you are familiar with installing scripts that require you to set up a MySQL database. If you aren't familiar or feel comfortable doing your own installations then you might want to go with Arvixe as Cim suggested since they offer Elgg installations.

While you didn't ask about support when it comes to Elgg, the Elgg community has been great at providing help and support where and when they can.

I would stick with one of the hosts form here: https://www.elgg.org/hosting.php

It's just one opinion but I would quit writing code before I would host on GoDaddy.

I agree, GoDaddy's UI and UX are just terrible and it's not cheap.

I did say:

If you aren't familiar or feel comfortable doing your own installations then you might want to go with Arvixe as Cim suggested since they offer Elgg installations.