They are not legal in all states. And they are designed for island sinks, not for regular installations where venting just might be difficult. Do a through the roof vent if at all possible. They do fail and will need to be replaced eventually. If you are doing an island sink, do an island loop vent instead. It takes up more room, but it's fool proof and failure proof. And it passes code in all jurisdictions.

That existing 4' high loop is really delving into a rather gray area of the code. Normally the trap is considered the official point of transition between an appliance drain line and the house DWV system and in that case the manufacturer of the appliance may determine the required diameter of the appliance drain line which is determined by the volume and velocity of the discharge from the appliance. Generally the manufacturer will print a table in the installation guide that defines the size, length and permissible vertical lift of the drain line with recommendations to increase the line size if you need to exceed the length or vertical lift. That table will also define a maximum length or lift which is determined by the output pressure of the pump in relation to pipe friction loss or maximum vertical static head. In most cases the code will require that the appliance drain must be in an accessible location to permit servicing in the event that line should become obstructed. Whenever a drain line is permanently installed in the wall it is then classified as a portion of the structure DWV system and it would require that there must be a trap at the beginning of the concealed section of the line and the minimum permissible diamter of the line would then be determined by the DFU (drainage fixture unit) load value of the appliance or fixture being served by that drain. In the case of a dishwasher the DFU load is 2DFU's which would require a minimum of an 1-1/2" drain line. The plumbing codes also require that a dishwasher, washing machine, HVAC unit or a home water treatment discharge line must also be connected by means of an "indirect waste receptor". Under some local codes you could achieve the indirect waste receptor by installing an "Air Gap" or they may require you to install a standpipe on the trap and connect the dishwasher drain to a standpipe in the same manner as a washing machine line is commonly connected to a standpipe. (Some codes require all dishwashers to have an air gap, regardless of how the drain is connected. Check your local code.) Under the UPC whenever we have two common vented gravity flow drains connecting to the same vertical riser we are required to use a "Back to Back Tee" (4-way) however code prohibits connecting two commonly vented drains by means of a Back-to-back if one of those drains is receiving the discharge from a powered drain I.E. a dishwasher or washing machine. The discharge from a powered drain is required to be connected by means of a directional fitting (baffled Tee) and generally the baffled tee should be placed lower than the gravity drain. In your case the simple solution would be to connect the dishwasher to the disposal input port, providing that is permitted by your local code as well. (For reasons known only to the AHJ some codes prohibit connecting a dishwasher to a disposal input port..One code that immediately comes to mind here is the Illinois State Code.) Your screen name is Tom-in-Seattle so I am assuming that you are actually in Seattle. I am not certain but i believe your code is modelled after the UPC (Uniform Plumbing Code). If so the defined trap and waste arm size for a kitchen sink is 1-1/2" but under the UPC all vertical risers and drain lines serving a kitchen drain must be a minimum of 2" pipe. You may use an 1-1/2" pipe for the vent. There is some question about whether the length of the horizontal offset on the vent in your attic would pass, but I doubt that it would even come into play unless you happen upon a hardnose inspector that is really intent on finding a problem.

Can use an air admittance valve if impractical to run a vent line for a kitchen sink. Cheap. Must be in an accessible location(not behind drywall) Valve opens when senses suction, otherwise closed. Generally have very long life but not permanent like passive vent line.

It appears that you have an S trap (no longer allowed) and it does not have a vent. A Studor vent would work to fix a vent problem, but you still have an S trap. The risk with an S trap is that suction from water going down the drain may pull the water out of the trap and leave you exposed to sewer gas. A P-trap with a vent (either a vertical pipe in the wall or a Studor) keeps this from happening. Bruce