Can a new player accomplish a 160k assassination mission without any upgrades to its Sidewinder

From my limited playing of E:D so far there's a couple of ways of getting improved profits per tonne, although ultimately the best way to do it, is work towards larger cargo modules and also larger ships.

The first option is just to work on finding the most profitable commodity to trade from a specific station. For that, the best way I've seen is to use something like the Elite Trading Tool, which allows you to enter a starting station and other variables and gives you the best routes that it currently knows about. These aren't always perfect, but generally I've found they're better than just using the in-game tools.

Another option is to trade "rare" goods. These are specific items which are available at some stations and which get considerably more valuable the further away you take them. You need a decent amount of cash to start doing this (to pay for fuel to get far enough away to increase the value) but you can make considerable profits with the right route (e.g. with a 40 Ton Cobra, I made ~500,000 each way on a run).

The trick with rares seems to be finding good clusters of items at either end of a run. For example this one

You can't. Currently, the class ranking of weapons doesn't seem to mean anything at all. Your choices of weapons are strictly limited by type, size, and targeting method. There aren't higher class small fixed pulse lasers, for example, there is only one variety of small fixed pulse laser.

What I think is going on here, is the game is ranking weapons based on their overall stats, regardless of power cost, size, or targeting method. For example, a huge plasma accelerator is class A, a medium railgun is class B, and a small fixed pulse laser is class F. There's not a class D small fixed pulse laser, because that's not how that class stat works. You can check out all possible weapons that you can find in the Elite Shipyard.