Can eyelid surgery make my eyes appear less deep-set and more forward in the eye socket?

These genetic variations are what make all of us different and unique.  This situation has no easy fixes, and I would caution you throwing a lot of money to something with at best modest results.

WIthout a photograph, what is generally meant by this is that the upper eyelids are hollow.  There is absolutely nothing reliable to address this situation which is a normal anatomic variation in the upper eyelids.  If this really bothers you, consider hiding the area with glasses that have a heavy frame.  If you don't need glasses, have plane lens with no correction placed in the frames.

a prominent brow with the appearance of deep set eyes is mutifactorial. see a plastic surgeon for your options. they range from simple camoflage with fillers to craniofacial orbit reconstruction. good luck

Without examining your situation it is hard to give advice. Often fat grafts to the upper orbit are successful in giving a youthful appearance. I would stay away from fillers as there is an increased risk of blindness following injections in this area.

I recommend a standard upper lid blepharoplasty and possibly a brow lift.  Be happy with this.

Tryint to manipulate your eyes is worse than playing with fire. 
 

Some people have deep set eyes which may or may not also have a "hollow" appearance.

There are many techniques to address these problems. The simpler solutions are often accomplished with fillers or fat grafting to soften the appearance of the eye, especially if it appears hollow.

In cases of very prominent cheekbones or forehead, other surgical options exist.

In all situations, the solution has to be individualized based on specific anatomy and individual preferences.