In PCB world, "fabrication" refers to making the printed circuit board and nothing more. "Assembly" means soldering on all the parts.
So, in your example, $100 will get you 29 pieces of copper-clad FR4, with traces etched and holes drilled. It is up to you to solder on the resistors, capacitors, etc which you need.
You'll typically want to panelize boards made for machine assembly. Loose boards are going to be more expensive to populate because special fixtures have to be manufactured and used (unless it's got really wide unused areas on the sides), and because there is more handling to populate (say) 12 single boards than a 4x3 panel.
That means more up-front NRE and tooling costs and more per-unit cost, which is pretty much the general result of not following guidelines.
You should put adequate tooling strips on the sides (typically 10mm bits that will be snapped off and discarded), tooling holes on the strips, and fiducial markings at least diagonally opposed on the tooling strips (for alignment), and diagonally about any high-density BGA footprints. You can see the fiducial marks (dots) on the tooling strips below.
Here is an example of using "mouse bites" and routed outline to allow such a PCB to be depanelized with a minimum of drama:
The board can be snapped along the lines with close-spaced (unplated) holes, leaving a smooth edge most of the way around. (the image shows a couple of ways of making the outline, but typically you'd need an additional routed line or V-groove to separate the boards in the other dimension).
I prefer V-grooves alone to mouse bites for volume boards, but it tends to cost a bit more, and they must extend all the way across the panel, which can be limiting. You have to pay attention to the stiffness of the panelized board assembly or it can cause problems in the assembly operation (so you can't cut too much away on a thin board).
Sometimes you can put test coupons for controlled impedance, wiring to power boards for testing or other disposable circuits on the tooling strips.
You'll want to minimize the setups and operations. A relatively small number of different SMT (only) parts only on one side of the board with adequate spacing should be cheaper for machine assembly. Anything that cannot be supplied in a tape and reel with a proper leader will cost more to assemble.
Minimize the number of different parts on the board (there are only so many part feeders on a P&P machine- 25 to 40, perhaps). It may be better to use a few more of the same part or otherwise rationalize your use of parts (if you need a 4K7 resistor for an analog circuit, maybe pick that for all your pullups, if you are using a zillion 100nF/10V ceramic bypass capacitors, try to to use them in other locations.
Much more could be said, and the illustrious Australian Dave Jones has some good videos on the subject- he was a professional PCB designer and has worked for Altium, so they're quite good if you use Altium. (often you can get the link from the right sidebar in SE) Altium has good built-in tools for panelization but, of course, you have to know what dimensions you want to use for the panel, spacing, route widths, allowance and copper-pull-back around V-grooves, fiduciary marking dimensions etc.).
Some of this varies from manufacturer to manufacturer, but there is a lot of common ground. Don't make the panel too big to fit in the assembly machine(!), something around 9 x 12" or thereabouts is usually fine for modest size individual boards.
The board itself should have adequate spacing between the parts, preferably no parts hidden under other parts or on the bottom of the board, proper pad dimensions, and unplated holes where desirable. Slots should be used instead of huge holes if through-hole parts have flat leads (eg. barrel connectors). Hole sizes for through- hole parts must have adequate clearance to account for tolerances or they can't be reliably machine-stuffed. Bigger is better (up until the point where soldering is compromised). If you need to cover bits for wave soldering (so holes won't get filled in, or connector tabs covered with solder), you need an additional masking or taping step. Parts that require alignment (through-hole LEDs, for example) can require custom jigs or fixtures, for which you might have to make allowances on the PCB.
Best Answer
The real problem here is the supplier, not 0402s going away. There are plenty of companies that make, and will continue to make, 0402 and larger resistor packages. There are also plenty of assembly houses that can handle, and will continue to handle, 0402 and larger. Resistors in 0805 package are still quite common. You've apparently found one, but I haven't run into a vendor nor assembly house that can't or won't handle them.
Switching technologies because this one supplier is dropping something is the wrong reason. Switch if you need the small size for your purposes. Otherwise find a real supplier.
As for the reliability, that depends on the assembly house. There is no reason 0402 and 0201 resistors can't work perfectly well if the assembly house is competent. Pretty much anyone can slap 0805 parts on a board. Starting at 0402, the process needs to be better controlled to avoid failures, but that is certainly doable. You should give the assembly house a tester, and they only deliver you units that pass the test. Failures are on them.
At 0402 and below, you also have to think about temperature balancing between the two pads of the part to avoid tombstoning. Reliability isn't all just someone else's problem.
Another issue is the power dissipation. With 0805 parts, you can largely dismiss power dissipation limits for ordinary signal circuits. For example, 5 V across a 1 kΩ resistor causes 25 mW. That's nothing for a 0805 package, which can usually handle 125 mW. It starts getting significant for tiny packages. When using really small parts, you have to think about power dissipation in cases you might be used to just dismissing it out of hand.
However, none of this should keep you from using 0201 when there is good reason for them. One supplier of the many many out there dropping 0402s is not a good reason.