Track bar joints worn out.
replace front brake rotors and pads.
1. out of balance wheels 2. misaligned front end 3. worn out steering dampner.
There are several causes of this but the most common we call a head shake. This is when the tires shake side to side, the cure is to insure the steering is tight and you add one or two hydraulic steering dampers.
Check into a new steering dampner, it's a common cause for shake at speed in a Jeep, but it could also be alignment or balance.
have your front brakes checked
Yes, it is possible. Any loose/worn front end part can cause steering wheel to shake.
No it is not common for dogs with beagle in them to shake violently. Some reason why dogs shake violently are seizures, and poisoning. If you dog begins shaking violently, it is best to take it to a qualified veterinarian who can figure out what is going on.
Worn steering and/or suspension components. Don't drive that fast until you can have the problem repaired, and try to NOT drive very far.
The transmission is not shifting in and out of gear or your front joint is going bad
Replace your steering dampner. Mine did the same thing.
More than likely tire out of balance or bent wheel. Have your tires rotated and balanced and all your suspension and steering parts inspected.
You need a steering stabalizer and if you have one it may need to be replaced.