Welcome to CloudyNights!
I'm not an expert (and there are many more knowledgeable people here than me), but I will give you what I can to answer your questions. more will chime in soon, no doubt.
personally, I am unfamiliar with the LX200 so I can't help you with the threaded rod thingy.
for image shift, this is something of a bane on SCT telescopes. in the SCT, focus is achieved by moving the primary up and down on the baffle tube. in order to do this, there has to be some mechanical clearance between the O.D. of the baffle and the I.D. of the mirror carrier. ideally, the clearance would be so small that the mirror wouldn't shift slightly as it moves back and forth in response to the pressure exerted by focusing. but these are commercial scopes and manufacturing tolerances are a fact of life in Mass Production so it's quite typical for there to be enough tolerance between the O.D. and the I.D. that the mirror 'wobbles' enough to make a noticeable shift in the position of the object being viewed.
that said, there are some solutions, some more effective than others. first, I would suggest running the primary through the full range of the focuser a few times. this has the effect of spreading the grease used to make the movement smooth over the entire length of the baffle. sometimes this is all that is needed to fix the image shift. other remedies include taking the scope apart, cleaning the old grease off and replacing it with new grease. still others involve more complicated modifications to the focusing mechanism.
most SCT owners just live with it.
I don't know if there is a 'best way' to store the scope; as in vertical or horizontal, mine live on the floor of my office in horizontal position because that's most convenient for me.
soft focus? well, there are some people who think that is characteristic of SCT's but I don't. I have looked through plenty of SCT's that are 'soft', but none of mine are. I attribute that to a couple things: first, you ALWAYS need to use a dew shield to prevent incident light from entering the tube and bouncing around. the corrector amplifies this problem, an the inside of the OTA is (typically) pretty poorly controlled for eliminating stray light. my SCT's all have been flocked on the inside of the tube and the inside of the baffle. another cause of 'soft' images are tube currents inside the OTA. there are lots of metal surfaces that tend to cool the air they are in contact with and cause turbulence in the double pass light path. making sure you acclimate the scope AND insulate it will sharpen your images up quite a bit.
SCT's can use the same barlows as anything else, in fact, because of the long-ish focal length (f10), even inexpensive eyepieces and barlows will provide good images. nothing special required, unless you have money burning a hole in your wallet
as I said previously, I am not familiar with this scope/mount, so can't offer any advice on upgrading the mount.
run the focuser back and forth a few time for the entire length of travel, get a dewshield and use it religiously and insulate the tube. you will see improvement with these simple mods.
and report back!!