Adding a vote for GC10. I've used probably 5 other LF pastes from Chip-Quik and Kester, none work as well as GC10 for me.
I think that type 6 paste is overkill for 0201. Type 5 or even Type 4 ought to give acceptable results. Finer solder paste also seeps more readily under the stencil if the gasketing is poor. Keep in mind as you decrease the size of the solder balls you increase the surface area/volume ratio of the metal in the paste. You expose more surface area to the flux and speed up chemical processes that you don't want happening. This is primarily why pastes "go bad."
I'd put more thought into your stencil and your apertures. The $15 add-on stencils that you get from many Chinese fabs won't cut it. I get my stencils from Stencils Unlimited which is more expensive but hassle-free, and they're quite good. They offer more advanced stencils with electropolishing and nanocoatings and etc. that make more sense the smaller you go. I used for manually stenciling a 176-pin BGA with nothing more than a flat stencil and a strip of masking tape, with perfect results. Using the same method I've also done micro BGAs, CSPs, and 0.3mm pitch DFNs with no major issues.
For apertures, I've recently modified all of my Altium libraries to use rounded corners for all pads and apertures. It's helped with paste release. Since you're printing manually your technique is extremely important too. I prefer metal squeegees, the rubbery ones I've tried never work out as well. Get a nice "roll" of paste going and get a nice, even stroke going with firm but even pressure. I find it easier to feel this out with a metal squeegee.
There seems to be no limit on how small apertures get for these parts. Sometimes when I'm watching random PCBA videos in production lines I'll see small passives with only half the pad covered in paste. Wild.
Understencil wipes are also crucial the
moment you notice imperfect gasketing. For me this is about every 5 manual prints. Squirt some 99% IPA on the wipe and give it a good rub on the underside of the stencil. For parts (thus apertures) that small I strongly recommend getting some clean-room wipes.
Proper cleanroom wipes, not just the green-box Kimtech wipes. Cleanroom wipes are designed not to shed ANY dust or lint. The cleanroom stuff is expensive, but it sure beats digging out tiny fibers left in tiny apertures by "lint-free" wipes. Another reason for rounded apertures IMHO - fewer pointy corners to grab those little fibers. Compressed air (or air duster cans) after the wipe help too, but only blow from the same side you wiped. If you go back and forth you'll more readily trap stuff inside the apertures. Generally speaking though I've found that using cleanroom wipes obviates the need for compressed air.