What is safer than a carotid endarterectomy nowadays, and where is the best center to do it?

Do your homework. Your local providers are likely excellent. Take a trusted person with you and ask questions. There are many types of providers, some who perform both the carotid stent and surgical endarterectomy. Stents have about 5% peri op stroke rate. Surgical endarterectomy have about 3% peri op stroke rate. Remember to ask what is your stroke risk absent any intervention.
Nothing. Carotid endarterectomy is very safe in experienced hands that is why it is important to go to a center with a large volume. A majority of the stent/endarterectomy trials for carotid disease favor surgery, except the crest trial.
Depends. On the patients age and other medical problems they have, plaque features and location, as well as previous surgery or radiation. Carotid stenting with distal protection is safe. Find out stroke rates of the doctor doing the procedures.
Very safe. Endarterectomy is the gold standard for symptomatic carotid disease. It is safe and effective. Many surgeons have a 2% peri operative rate of stroke, less than 1% risk of peri operative death. There is some risk to nerves that control the voice box but should an injury occur it is usually temporary. The key is to ask your surgeon what his personal complication rates are.