Do you think the T245 can work on 0402s, though? And would a T245 iron be able to work under a microscope?
Yes and Yes with the right tip.
It seems like there's general components that work on most or all PCBs, and IC's that are specific for that hardware that contain software? Is that how it works?
Passives would be things such as resistors, capacitors, and diodes. There are also active components, such as transistors, voltage regulators, ... sorts of things that are common. IC's fall into the latter category as well. These can be referred to as jellybean parts, particularly those that don't require any code. Just wire it up properly to make the circuit work.
Then you start getting into user-configurable/programmable IC's (i.e. contain a ROM or are tied to one) that store the necessary code to make it work. Still common off the shelf components (COTS), so anyone can buy them without licensing/IP violation issues. FPGA's and micro controllers would be such examples. Just have to create the hardware design and code needed to make it work as intended (makes them very flexible).
There are also custom IC's, which are designed for or by a specific manufacturer that they have made for them (or in some cases, can also manufacture the chips themselves in-house, such as Samsung). These are specifically designed for the products they're used in, and not made available to competing companies or general public. Apple's A6 chip for example, is based on an ARM processor design they've licensed, but has extra stuff added to make it an ideal component for the iPhone and iPad tablets they're used in. Cuts down on the number of separate components, which can save on costs, can reduce power consumption, and take less PCB real estate for the completed design (i.e. potentially allows for a smaller product). This is viable due to the number of units produced, as it keeps the per unit cost within an acceptable level. It also makes a device that contains them a PITA to fix compared to designs that are based on jellybean parts, as it's much more difficult to get the parts or the code from any source other than a working donor device (at least the part you need).
Hopefully this is enough to clear at least some of the confusion (tried to keep it as short and simple as possible).