Looking for feedback on my microcontroller-controlled boost converter
Start-up currents can be huge, and need to be managed.  A converter would have a current sense of some sort.  You'll want to either sense the coil current and limit it, or try to deduce it from the rate that the output cap is charging and, again, limit it.
Simulate.  I think you'll find that the transistor won't turn off very well.  


You need to saturate it pretty thoroughly to make it turn on, to turn it off quickly you need to actively pull the charge out of the base.  This requires some hefty drive.
(Zetex had some app notes on using BJTs in switchers about 20 years ago -- they were trying to sell their super-beta transistors for the task.  I built a circuit to try it out and it worked well, but in the end it's as complicated as using FETs).
(Zetex either became, or was bought by, Diodes Inc.  They have some pretty impressive BJTs)

To drive 200mA out, the average current in the inductor needs to be 800mA.  


With an ordinary BJT this implies 80mA to the base, which is beyond your typical micro pin (and well beyond what you'll get with a \$1000\Omega\$ base resistor).  
I'm not sure how things pencil out with a super-beta transistor, but even if the idling base current is less, you still need to actively drive current into and out of the base.  Essentially, you end up with a gate driver, followed by a typical resistor-cap-diode gate speed-up circuit.  It'll make you ask "why didn't I just use a FET?"
1MHz, 800mA average coil current, 75% duty cycle -- that says that at the edge of continuous conduction you'll be charging from 0 to 1.6A in 750\$\mu\mathrm{s}\$.  
That says 2.2\$\mu\mathrm{H}\$ to me, unless you want to insure continuous conduction over a wider current range.  
10\$\mu\mathrm{H}\$ isn't a bad place to start, but it's more than you need.  OTOH, it slows down the circuit's response, and therefore eases the load on software.