How to package a 16 bit integer to send it with serial.write?
You need to send the uint16_t as the two the individual bytes it is composed of, one at a time, in whatever order your Arduino code and the code on the PC agree on: (MSB first, aka, big-endian; or LSB first, aka little-endian).

Since it would be possible for the two devices to get out of sync, you may also want to package up the two bytes with a third, or "start-byte". Pick some fixed value (such as 0xCC) that has a couple of its MSbits set, then the MSB, then the LSB. Your receiver can know to look for that fixed value, then the MSB (which it knows can't have those high bits set because the analog data is only 10-bits), then the LSB (which could have any value).

The PC should only accept a three-byte packet when it has received one that fits that scheme. Otherwise it waits for another 0xCC (or whatever value you choose), collects two more bytes, and tries again.