Before I started refinishing, I cut the opening a little bigger to accomodate my sewing machine. TIP ALERT: when cutting veneer mark the cut with a pencil and then score the veneer with a utility knife. Then use a jig saw with a very sharp blade to cut "inside" the score...if you don't, the veneer with split and splinter and you will make quite a mess!
Normally I use Formby's Furniture Refinisher to strip latex paint and old varnish...but two layers of paint needed a little more muscle....so I used a citrus stripper. I will not use anything stronger on veneer (always use any stripper per the instructions and in a VERY well ventilated area!) This piece didn't have a single chip in the veneer and I didn't want to risk hurting the piece by using industrial strength stripper! Took a little more time and more than one application but it was worth it.
After removing all the old paint, I took 0000 steel wool and mineral spirits and gave the wood a "bath." When stripping wood, regardless of what you use to strip it, always "wash" the piece with mineral spirits to remove all the "crumbs." After it dries, take VERY fine sand paper or 0000 steel wool and sand lightly before applying stain or a finish.
TIP ALERT...keep old toothbrushes on hand to get into groves and corners when stripping furniture. DIYers never throw away old toothbrushes, coffee cans, socks, old tshirts or towels...trust me, you will need them!