Names are not so useful, emails yes! Export their emails in Linkedin and then upload to Facebook "audience" targeting and then you're good to go:)

Yes and no!
While Facebook is more flexible and its users are more likely to respond to your direct approach, on Linked In is more difficult.
From my experience, on Linked In is more lucrative to share your problem/issue in a form of a question with details on group's board, where the dynamics of Q&A is much better.
Of course you could address your connections directly or by an email campaign, just don't be aggressive with the approach.
Most important (and the determinant for your addressing way and channel) is your product/service that you offer.
If you think I could be useful, you may reach me by email at bogdins at gmail dot com or by skype at technobogdins.

There are various ways to target specific people, although you need be be careful relative to the social networks' terms and conditions. If these people have opted in to your list, you can use Facebook customer audience retargeting to promote stories in their feeds. If they haven't opted into your list, you may have to find ways to reach out to them directly. There are lots of strategies for this, but it really depends on the specific reason you want to reach out to them.

Use the audience function in FB, you need at least 100 emails to build a campaign triggered only to those users when they are on facebook

Hi,

sadly, knowing the names of your customers is not enough. However, if you have their email addresses or phone numbers, it is possible to use the so-called Facebook Custom Audiences and create a custom list of those people.

You would upload the list of email addresses or phone numbers to Facebook and it will anonymously match them to individual Facebook profiles - after that happens, you will have a custom target audience that you can reach with any ad campaigns. This can be very powerful, as you can reach exactly just your current clients or subscribers and create highly-targeted advertising campaigns.

If you had such information (emails or phone numbers) and needed any help with creating Facebook custom audiences, feel free to get in touch and I'll be glad to help.

Cheers!
Adomas