yes
Your tires need to be "high speed" balanced.
could be struts and or strut brg plate that mounts the strut at the top
No it will not cheak your tie rods .I'f it vibrate's when you press your brake's it's the rotors, fix it quick or the "vibrate" will keep losing lug nuts. Yes.
bad engine mounts low rpm old timing belt over charged ac
We have one that does that. The mechanic told us it was 2 bad motor mounts, but I haven't fixed them yet (so cannot verify the diagnosis)
This is probably due to worn brake rotors on the brakes, if there is no vibration when driving then this will be the cause.
The power steering system works on high pressure and at idle the pump does not turn fast enough to provide full assist. It requires a higher engine rpm.
If other things vibrate as well as the steering wheel you might have a damaged motor/transmission mount.
There are several things... assuming you mean while the vehicle is in motion, possibilities include worn/loose steering components, out-of-round tires, excess toe in or toe out in the steer axle, out-of-round brake drum(s), bad engine mounts... choose your poison.
Muscle spasms can cause this.
check motor and transmission mounts first. check steering and suspension parts.
Steering fluid leak.