spoon- for the bars, bullhorns are a popular choice and seem to be accepted more than flat bars or risers when put on track bikes. Keep the beautiful track drops the bike came with, and put on some Nitto RB-021 or RB-019 (or whatever they're called).. alternatively many people take old road drops, flip them upside down and cut off the drops so you get bullhorns.. nice cheap option, many hand positions. Try the bike out first, you may like riding in the track drops... but many aren't comfortable with this for city riding.

As for brakes.. with bullhorns many people run the bar-end time trial levers, which look real slick and if you ride on the horns a lot, they're right there. Otherwise, people run "cross" levers, which use the same motion as mountain bike levers, but are much more tared-down.. mount right next to the stem. You could probably use your old brake calipers, but check the pads.

Wheels.. luckily most clinchers prebuilt for fixed-gear/track bikes are incredibly durable at best. Look for formula hubs, which aren't anything special, but hold up well under most conditions. Velocity Deep-V's are a favorite around here, but they're heavy and somewhat expensive. Weinnemans or Mavic CXP-22's are also a good bet. You can get a hand built wheelset on ebay for between 120-160 that should suit your purpose.. not incredibly lightweight, but reasonably durable. Just search for "fixed gear wheelset". Tires.. the toughest are gatorskins and armadillos, but rubino pros are fairly common as well.

All helmets certified by CPSC or whatever that organization is called are just as safe as others.. you just get more ventilation and lighter weight, and maybe a better fit with more expensive helmets. Try them on before buying.