Can I install both "Basic" and "Premier" products on my desktop in order to prepare more than 5 efiled returns for family?

No. The IRS limit is 5 efiled returns. You can't increase the number with multiple installations.

No that won't work.  And you can only have 1 program installed.  5 is an IRS limit.  If you need to efile more you have to buy Intuit Products for Tax Professionals https://taxpro.intuit.com  and sign up with the IRS as an ERO - an Electronic Returns Originator.  How to become an Authorized E-file Provider…..

http://www.irs.gov/Tax-Professionals/e-File-Providers-&-Partners/Become-an-Authorized-e-file-Provider

Maybe some can use the Online version? Like the Free Edition .

perhaps some qualify for the free file online. 

this is from the iRS

https://www.irs.gov/filing/free-file-do-your-federal-taxes-for-free