Help analyzing SDS-Page gel
Here are some thoughts based on my own experiences purifying many proteins and running many SDS-PAGE gels:

(1) Resolution--how fast did you run this gel? Often if you run at higher than say 150V or so you can get bands that look like this. The resolution could also be related to..

(2) I would bet the streaking is due to salt in the sample--a lot of the time lysis/wash/elution buffers can have upwards of 500 mM or 1 M salt, and this invariably causes streaking. Load less sample in these lanes and keep the total volume the same to dilute the salt below 250 mM at least. Other things in your buffers could be causing the streaking, but salt is the most common (let us know the composition of your lysis/wash/elution buffers).

(3) Your purification of GST alone looks very good--often your preps will look much more like the lanes you have indicated, which are not entirely clean. This is quite common for a one step purification--often you need 2-3 for certain proteins to completely remove contaminants. There are several possible sources of these bands:


Endogenous proteins: Many bacterial proteins will non-specifically stick to the beads (this is why we wash), but even thorough washing can still leave some contaminants. These proteins also could be bound to your protein of interest and are co-eluted off of the resin.
Truncation products: Depending where your tag is (let's sat it is on the N-terminus), you can sometimes get incomplete translation products--they have the tag, but they are not full length proteins. These will still bind and elute from the beads because they have retained the tag. Ignore this point if you tag is on the C-terminus, as in that case the tag will only be on full-length protein. 
Degradation products: This is related to your later questions--often during the purification your protein will be degraded by bacterial proteases. Did you include a variety of protease inhibitors in your lysis buffer? Keeping all of your samples on ice is also important for this reason. 


(4) Understand that degradation products do not mean you get only your protein and the fused GST--they can be a variety of truncations due to degradation by proteases. 

(5) I would bet this is just due to your running conditions or the salt in your samples. I think this is probably actually your single GST band.

(6) Glutathione will not show up on the gel--it's formula weight is about 300 g/mol if I recall meaning it's about 300 Daltons, so much smaller than anything that you'll retain on the gel. Further, remember than your stain is for proteins--Coomassie blue binds basic residues on proteins. 

(7) I'm not sure which band you are referring to, but I think the very bottom band in your marker lane is simply the dye front. It could also be a degradation product of a protein in your ladder.

As far as the second gel goes--it is all due to the buffer level falling during the run. Without buffer covering the top of the gel there is no current flowing. Probably as the leak was occurring you had uneven distribution of the buffer. Strange things can happen when this occurs (unscientific, I know).