1. Is it possible to set up a split tunnel vpn so that only traffic going to SoftLayer's private network will go through the vpn tunnel, and all other network traffic will use the default internet connection.
Yes, it is.
2. If this is possible, will it affect inbound traffic to the server (It is a hosting server).
No, it won't.
3. Are there any recommended applications for a linux server to do this split tunnel vpn?
I suppose openvpn would be the easiest setup to implement.
4. How challenging is it to install/configure such a setup?
Not too challenging if you have some experience with openVPN. I'd estimate it'd take me around 2-3 hours to set things up and test functionality.