Currently no. There are known issues with the graphics drivers and glitches in the software as a result. You can get SL to install, but the performance and stability are lacklustre at best. And Apple is unlikely to add driver support for SL in future revisions of that operating system.
Additionally, you cannot manually add drivers for core components to OS X. You may be able to modify kernel extensions, but not at the level required to get the old OS operational. You may be able to port the graphics driver from Lion, but that is certainly a long-shot.
You can follow a discussion over the the Apple Forums for additional information, where it has been reported that the new Air installs a special build of Lion that carries with it a unique graphics driver. One user claims they got SL installed, but it does not recognize the internal display. That's a pretty big show-stopper, unless you want your new Air to be a desktop.
Your trip the Genius bar was too focused in my opinion. I suggest you make another appointment and lay out your case above, but this time, don't mention the memory tests, but that the problems showed up when you upgraded the RAM. Refer them to your old case number (tack this visit on as a follow-up).
Tell the Genius on hand that the problems still exist and that they need to do further testing. As it stands, your Mac is unusable. Additionally, tell them to make note that you have tried a fresh re-installation and it did nothing to help (don't mention that things ran fine for a day or too, stick to the current state of the system), which will rule out any software problems.
Ask them on what they suggest and follow it (no matter how annoying it may be). If it doesn't fix it, call them and do another follow-up. If they tested the RAM and it passes, then it might be the logicboard.
In any case, do not visit them with "the solution." Let them find it through there own testing. Clearly, the logs containing kernel panics and all the problems you list will support your case. Moreover, their internal testing tools are much more sophisticated than what you'll scrounge up on the net through 3rd party apps (they actually have in-house diagnostic utilities for these occasions).
I would highly recommend that you stop spending your time diagnosing the problem and let them do it for you. That's what they're there for.
If after everything is said and done, and they still can't pinpoint a problem, you will likely walk away with a new Mac. Just focus on the problems you are suffering from and leave the diagnostics to the Geniuses and technicians.
You may want to downgrade back to the 4 GB for now. And it's of note that replacing the RAM and the hard drive yourself do not void your warranty for that particular Mac model.
PS: You don't specify if you took your notebook to an Apple Store and spoke with a Genius or some Apple authorized repair centre. I would strongly encourage you to deal with Apple directly. They'll not only treat you better, but likely solve the problem with less headache. My advice is tailored to them; you are totally on your own should you choose a 3rd party repair shop. To galvanize the need to deal with Apple, let me put forth the following. I had a friend who purchased a Mac mini. There were strange problems during boot up where it would occasionally load the keyboard set-up assistant. The machine was also sluggish and exhibited other unusual anomalies in its day to day. We took it into an Apple Store and the Genius tried to reproduce the errors. He couldn't. He also expressly stated that minis never required repair or servicing. But with all that, he elected to replace the entire unit and gave my friend a new mini. Took it back to his place and it hummed, no more issues. I doubt he would have received the same treatment from a computer store authorized to service Macs. Deal with Apple direct; I cannot stress that enough.
Best Answer
Based on our discussion and some fact finding above:
I would say your hardware is fine /the Intel HD3000, but it could be on its way out!, means it could be dying the slow death.
You can double check and run a stress test for the GPU using this little program, or look for your OSX version.
GpuTest 0.6.1 is a bugfix release for Mac OS X only. The CSV file (for benchmark scores) was no longer generated at the end of a benchmark. This CSV file is stored in the root folder of the current user (_geeks3d_gputest_scores.csv).
However, I still think you have a software (programs) interference issue. Remember the Intel GPU uses shared memory, so that could lead to a problem you are having.
Finding out which program is interfering could be tricky, but you had to just turn them completely off one at the time.
In your case I recommend:
Use an external monitor (or LCD Television) to:
1-see if it is a problem with the build in display
2a problem with the flat band cable to the build in display
3-actual problem with the GPU chip