Let's assume my LinkedIn account shows the name of my current employer (as most accounts do). Can the employer ask LinkedIn to delete posts of mine which they don't want to see associated with the company? Can they do this even if the posts are by no means offending or inappropriate? Does this right differ by country?
They can certainly ask. Whether or not LinkedIn will comply, only LinkedIn can say for sure. Or waiting to see.
But, since your employer objects to these posts, do you really want to upset them?
Remove the posts yourself, apologizing is not necessary, but would not hurt.
Then ask you employer for a clear, written policy on what you may post when their name is visible and will appear to be associated with it.
Absent legal weight behind the request or breaking of Terms Of Service, it is unlikely that LinkedIn will simply delete the content of other people at someone else's request.
Deleting something simply because another company doesn't like it will ultimately have a chilling effect on the rest of the site userbase. This can get people to leave in droves and promote the use of competitors. As such, LinkedIn doesn't even do this for Microsoft, the current owners of LinkedIn.
The employer can potentially get it deleted under a legal objection, but LinkedIn is no YouTube in that not everything is 100% automated and customer services do exist, so such a legal challenge is not automatically guaranteed and may need actual weight behind it beyond a claim.
They can try...
In practice, it'll be difficult and your employer would need to provide a very good reason for wanting your content to be edited.
LinkedIn's editing policy will be governed by LinkedIn, so you'll need to ask them directly, or use the official support forum to ask this question.