Not sure what e7 type means??
As far as oil, going by my Autometer oil pressure gauge in my 2004 with sender connected to to top of oil filter housing, I have been running
5w-40, viscosity with temperature below 50 F (10 C)
IMO, took too long on cold start up to get oil pressure up. Then 15w-40 in warmer 50F+.
About half way thru 10k OCI use STP Oil treatment (german invention :wink2

for zinc boost.
Should have real Oil pressure gauge OEM is fake, choose weight by your environment.
Hopefully compression good!! Been issues with rings.
06 has piston ring gap for emissions. see
https://www.cumminsforum.com/forum/3rd-gen-powertrain/1323945-piston-ring-gap.html
Dodge /Cummins compression test info.
CYLINDER COMPRESSION PRESSURE
Ensure batteries are completely charged and the engine starter motor is in good operating condition. Otherwise, the indicated compression pressures may not be valid for diagnostic purposes.
1. Disconnect the fuel inlet line to the fuel filter housing. Plug the fuel line from the fuel tank.
NOTE: Failure to plug fuel line will result in fuel leak.
2. Remove fuel transfer pump fuse/relay.
3. Start the engine and idle until the engine stalls (runs out of fuel).
4. Remove the cylinder head cover. (Refer to 9 - ENGINE/CYLINDER HEAD/CYLINDER HEAD COVER(S) -
REMOVAL)
5. Remove the cylinder head cover carrier gasket. (Refer to 9 - ENGINE/CYLINDER HEAD/CYLINDER HEAD COVER(S) - REMOVAL)
6. Remove the high pressure fuel line between the cylinder head and fuel rail for the cylinder to be tested. Use tool# 9011 to cap this fuel rail on the cylinder being tested.
7. Remove the fuel connector tube nut and fuel connector tube.
8. Remove the exhaust rocker lever.
9. Use Tool 9010 to remove the injector and copper sealing washer.
10. Install the exhaust rocker lever and torque to 36 N·m (27 ft. lbs.).
11. Cover the remaining rocker levers with clean shop towels to prevent any oil splatter under the
hood.
12. Place a rag over the compression test tool fitting. Crank the engine for 2–3 seconds to purge any fuel that may have drained into the cylinder when the injector was removed.
13. Connect the compression test gauge.
14. Crank the engine for 5 seconds and record the pressure reading. Repeat this step three times and calculate the average of the three readings.
NOTE: The minimum cylinder pressure is 350 psi. Cylinder pressure should be within 20% from cylinder to cylinder.
15. Combustion pressure leakage can be checked if cylinder pressure is below the specification. Perform the leakage test procedure on each cylinder according to the tester manufacturer instructions.
16. Upon completion of the test check and erase any engine related fault codes.