It depends on the doctor's office that you go to, what technology they have available to them, and what procedures they are comfortable with. Sounds like you are a perfect candidate for a combined Active/ Deep FX treatment. It is not a peel, but a fractional CO2 laser resurfacing treatment. It will probably give you less downtime and better results than an average chemical peel, but it will likely be more expensive. Some offices don't have this laser, and they might want to do a chemical peel, usually a medium depth TCA peel, or perhaps use a similar laser.
The veins might not improve with either of these treatment as well, but sun damage, eye and lip wrinkles will likely get much better. Any procedure comes with risks. In general, the more damage you have, the more intensive of a procedure you will need, with perhaps higher risks.
The risks are minimal in the hands of an experienced provider. Your doctor should discuss your options at the consultation, and go through risks and benefits of each procedure.
