My steering wheel on Merc C280 1995 model. While driving I can move the steering wheel slightly left to right, but the car remains straight. Then the car will veere right and then left slightly without me turning the wheel. I have to rectify it

Well the next thing to check is the wheel bearings and adjustment

sounds like the gear box or sterring rack went out have some one move the wheel look see what is moving and wha tis not

2 ways to resolve. Take steering wheel off and recenter on spline ( done by dealer or auto electrician due to air bag) , or do a wheel alignment adjusting tie rod ends same distance as each other ie move both wheels left to right or vice versa.

Try turning the key while jerking the wheel left and right and depressing the brake pedal. If that won't work try another key.

What seems to be the problem with your turning signal cancellation, is the steering wheel is not in line with your wheels.
Meaning, when driving straight, take note on on where the steering wheel is positioned. If the spokes or logo on the steering wheel are turned or not level, then that's the reason for your turn signal is doing what it does.
The best way to cure this problem is take your vehicle to a wheel alignment shop that you trust, and tell the technition to position the steering wheel straight with the wheels.

It sounds like a worn steering rack bush in the steering rack,as this explains the play youve mentioned,the vibration is caused by the wheels moving very slightly through the play in the rack bush,theres 1 letf and 1 right side bush in the ends of the steering rack that would need checking for play or movement

Wheel bearing gone.

Drive slowly down the middle of a straight road, let the steering wheel go and apply the brakes. If the car wants to turn by itself check your ball joints and shocks. If the car remains straight you may have an irregular wearing tire which causes it wonder at higher speeds.
I hope this helps
Sam

Check both front wheels with the steering wheel centered. One or both may appear to be pointing in the wrong direction slightly. Check both sleeves on the tie rod ends to make sure they are tight and not able to turn. (adjustment sleeves are slightly different than on most cars. One sleeve extends from the drivers side across the front and is about 21/2' long the other is shorter and is about 5" to the passenger side of the pitman arm on the steering box. If either moves, turn it 'till the wheels are fairly straight, tighten the bolts that are on the clamp around the slotted tube, and bring it to an alignment shop and have the toe in re-set.
A condition such as this is nothing to mess with. loss of steering is extremely dangerous.

Get this vehicle into an alignment shop and have the front end checked out before you kill someone