At 5'10/34" inseam, your probably going to want a bike with a full-size frame...WR250R or KLX250S are better choices than the other small bikes mentioned.
That said, be aware that even a DR650SE thumper is as light as some multi-cylinder 250cc streetbikes, and not a whole lot more powerful until it's uncorked. It's not like riding a 45+HP race-ready SM with a hard hit of power. It's a bit more forgiving. Where something like a DR650 will often seem too big and powerful is when riding technical dirt as a noob. I wouldn't really want to start out on anything this heavy in technical dirt as a kid with his very first moto, but it is a quite easy bike for an adult of your size and training to begin to ride on pavement first, and then transition into easy trails. Stripped of 20lb from the common swaps of heavy muffler, battery, rear underfender, mirrors, and metal tank, the DR is only around 40lb heavier than the 250s, and even more weight can be shaved off (handles, helmet lock, passenger pegs, huge taillight, etc.). If you had an easy time of passing the BRC, you may not have any issue with a mild-powered and sub-400lb 650 thumper as a first streetbike. The Rider's Edge BRC uses 500cc Buell Blast thumpers that aren't really any easier to handle than a DR.
As for gear, if the helmet is DOT and/or Snell approved, the protection is similar. Does it fit YOU comfortably? Extra price after DOT/Snell is brand, lightness, quietness, features, and luxury. I wouldn't spend more than $150 for a helmet. My $70 Vega Attitude is noticeably not as cushy as a $500 Arai, but it has the same DOT approval and is comfortable enough for me. You have me wondering though, why you want to wear a hi-viz vest over a jacket, rather than just wear a hi-viz jacket? Check out TourMaster Flex pants and something like a hi-viz Tourmaster Sonora or Intake mesh, or Icon Hooligan 2 Mil-Spec hi-viz mesh jacket. Summer gets hot without mesh, and these hi-viz jackets all have insulated liners for cooler weather too. The 3/4-length touring-cut Sonora will also accept chest armor in front pockets, like some of the pricier Alpinestars jackets do.