Learn English – What does “I feel for that” mean

Dragging someone through the mud implies attacking their character and motivations, or accusing them of wrong-doing. Mud is a metaphor for evil or wrong-doing. A related term is mud-slinging, which would be throwing mud at someone.

Depending on the exact quote and the context, the speaker could be saying that his accusers can accuse him of anything, but that they should refrain from accusing his family. Or the speaker might be asking that accusations be made in private and not in public where his family would have to hear about it and would also be tarnished by it.

This sounds like a quote of a politician who is basically saying that politics is dirty and (false or exaggerated) accusations are often made, but politicians' families should not be dragged into the fighting.

A "hue and cry" was originally a loud shout after a criminal ("Stop Thief!") to alert others nearby to chase after them and catch them.

Now, it usually means a public ruckus about something that may or may not be criminal or illegitimate. There could be a "hue and cry" about the AC making the building too cold for example. It could even just be a loud gossip and pity party that doesn't really mean to be productive or solve the problem the members are complaining about.