Without knowing when the last time you had your transmission serviced I would say getting that done now would be $100.00 well spent .
Straight 6, 4 cylinder huh? Maybe a 4.0 liter. The tranny needs to be checked out.
Go to www.atra.com and find an ATRA certified shop near you. Do not go to a transmission store like AAMCO.
The hose is the PCV valve and associated hoses.
The problem is likely the transmission torque converter.
That circled part is the PCV vavle (or crankcase breather).
It has a vacuum, and sucks the gasses from the crankcase, to avoid to much pressure building up.
"Also, when I come to a stop or slow down at a light, sometimes my car will shake really bad off and on"
IF this PCV / Crankcase Breather is loose, split or broken. It will cause an idle problem, which result in a shaky car when at a stop. Simply because there is a vacuum / intake leak accuring. Replacing the PCV vavle is a simple DIY job.
My car is an automatic straight 6, 4 cylinder jeep. I've been having what I think are transmission problems but I'm not sure. Occasionally, not too often, my transmission does slip, I'll be giving it gas but it won't move, and then all of a sudden it will jerk forward and start going. Also, when I come to a stop or slow down at a light, sometimes my car will shake really bad off and on, and other times it completely shuts off, and when I try to restart it, it takes about 5 minutes, the battery turns over, so I don't think it's that or my starter. I was looking at my engine today and noticed this tube, I'm not sure what it is, but it was completely broken, and the tape was barely holding it, I readjusted the tape but I'm hoping replacing this tube will help? Any advice or ideas on what's going on and what this tube is and it's purpose?