I use ZB3040H manual stencil printer for pasting PCBs. It uses 40x30 cm framed stencils, which is more than big enough for my PCBs and panels I work with. I typically order them at JLCPCB.
I might have to do some thinking on that. Googling that one specifically, gets me about $430. Surely the concept can be done for cheaper than that. Maybe not as nice, but...
For reflow I have ZB2520HL oven. It seems to work pretty well with boards as large as 20x20 cm, it's specs says it can work with PCBs up to 25x20 cm, but 20x20 cm is the largest panel I've actually reflowed. You've got to make sure you DO NOT place PCBs right on a drawer because it will wick away a lot of heat off PCB and will lead to solder failures. That's another reason I need mounting holes, so that I can use small M3 screws for standoffs just lift a PCB off the drawer and reduce heat transfer to a drawer and provide enough clearance for components on the bottom side during reflow of the second side components.
Oh, you have a REAL reflow oven! I got the cheapest toaster oven that I could find, on purpose: thin sheet steel all around, no insulation at all, the thermostat is a bi-metal switch outside the chamber, etc. I got the cheap one on purpose, so it would have all-mechanical controls that I could easily set out of the way. No smarts that can't be removed easily and would get horribly confused.
I set the thermostat all the way up and the timer to never shut off, and plugged it into a software-controlled triac outlet. That software takes feedback from a K-type thermocouple inside the chamber, to run a configurable profile made from ramps and holds, and then opens the door with an R/C hobby servo to cool faster. (my own design)
I've done double-sided boards too, by doing the "lighter" component side first, and then just setting it upside down on the bench and then the sandwich tray that came with the toaster oven. Sounds like you have a better set of tools than that.
Now I see why are you complaining that solder paste gets under the stencil - no wonder because boards with components on the underside is not going to lie flat
There is no need to "sandwitch" PCB for a second reflow, as surface tension of a molten solder will ensure parts stay in place. In my experience even large-ish parts like 2917 size tantalum caps stayed in place no problem.
The sandwich tray is a slightly corrugated metal thing that came with the toaster oven. You're supposed to put literal sandwiches on it. It also came with a wire rack that is supposed to be for other foods, or maybe in combination with the tray to catch drips or something like that. Anyway, it's dirt cheap, designed for food, and I've replaced the controls to use it for PCB's.
For the second side, I just put the first side face-down on the bench to place the parts, and then on the tray again to reflow. If it doesn't sit flat enough, I'll put a ring of blank PCB's under the edges. Haven't had a failure yet, except for a few tombstones and the QFN that got too much paste on the thermal pad. I often feel the need to clean up after the stencil though, so I probably do need something better than just taping it down to the bench.
I also have ATTEN SS-862D hot air gun (I think Louis Rossmann recommended it at some point, which is why I purchased it), and it's FANTASIC! It's got high enough power to quickly heat up parts for rework, This additional power is very helpful for 6+ layer boards with several ground/power planes as those planes can wick away quite a bit of heat, and so your hot air gun needs to be able to pump heat faster than it's wicking away so that solder around component(s) you are working on can actually reach melting point.
I have a cheap combination iron and hot air station, with a thermostat for each. Seems to be bulletproof though, and definitely well used.
(For more than just solder too: I left the hot air on a low setting for several hours trying to straighten a small rubber hose. Small nozzle stuck into the hose, and the handle supported with string. It...sorta worked. The air gun was fine, but the hose barely responded.)
I also developed a small C# program which takes gerbers of a PCB and a BOM file and generates an assembly document, which shows which parts go where (see attached pdf as example of what it looks like). As my boards tend to contain a lot of 0402 parts which do not have any markings on packages, making sure I don't place wrong parts is extremely important, which is why I invested my time in that program. Its' sources are available here: https://github.com/asmi84/GerberTool but it's very ad-hoc program which was designed around my process, so if someone else wants to use it, he/she will probably have to adapt it to their process. I also use that file (in printed form) as a checklist for gathering parts before assembly, as well as marking parts already placed. Before I begin assembly, I always gather all components required into one box, and have another box for leftover components after placement. This also helps to keep tabs on what's placed and what isn't. This slows down assembly a bit, but it helps to ensure correct parts are placed in correct places.
My bench has an LCD monitor on the back wall, driven by a Raspberry Pi. Its USB and GPIO are available on the bench, via a long-tailed hub and a 40-pin ribbon, and WiFi to my main network so I can put the KiCAD project on it while I place the parts. Seems to work well. It's also one of several machines that can run my USB oscilloscope. I specifically got a monitor that had a DC power input, so I could tap off of that to run a DC-DC converter to the Pi. One plug-in AC switch covers both...and I'm back to the "It's now safe to turn off your computer" mentality, for those that remember that.
It also helps to use as much of the same part as possible. (fewer lines in the BOM, with correspondingly larger Qty for each) Analog designs need a zoo of different values, but even then there's room to consolidate. Two "standard value" resistors in parallel, for example, instead of a single "half value", so I don't have to keep straight that "that one" is different. Digital designs, in my experience, are even more forgiving. If power is a non-issue, then a pull-up resistor can have several orders of magnitude tolerance.