garbage disposal relocation
Good pictures.  You have an unusual configuration for that drain.  Typically, there is only one drain line coming from the wall into the kitchen cabinet.  That would have one trap and upstream of the trap are the sink, disposal, and perhaps a dishwasher drain (you don't appear to have one in this cabinet.)
What you have is not necessarily wrong, but if you search for garbage disposal drain on Google Images you will see many examples of more typical configurations.  Yours could be like that if you re-orient the disposal and re-configure the drains to combine into one trap.
If it's easiest for you, you could use only the lower drain, and stop using the upper connection.  It seems like you need room for a deeper sink and that may help.
Why is your sink drain unusual?
The gotcha question is -- why is your drain setup unusual?  And why is that Air Admittance Valve (AAV) (not exactly a vent but it has a specific purpose) connected to the upper drain?  Does this drain lack a proper vent at all?  Check outside your house and see if there is a roof penetration for the vent above your sink or elsewhere in your kitchen area.  If not, that AAV -- which is not a total substitute for a vent -- will need to stay in there somewhere downstream of the trap -- but you could cut some drywall away and see if there's room to relocate it into the wall cavity (just don't cover it with new drywall after; you cannot cover up an AAV because they have a limited lifespan and need to be serviceable.)
FYI the Black &amp; Decker The Complete Guide to Plumbing book is great, well illustrated, and you can find it at many libraries.  I recommend you read it or another plumbing book so you can have a better understanding of traps &amp; venting.  That may help you understand your options for making changes to this drain better.