When troubleshooting rear derailleur setup, there are some basic steps. (I know you said you've checked some of these. I'm still including them for the sake of a complete answer.)

1. Disconnect the cable from the cable fixing bolt, and remove it from the derailleur.
2. Remove the derailleur from the frame.
3. Check the alignment of the derailleur hanger. Many shifting issues start there. Even a new derailleur hanger often needs to be aligned.
4. Once your hanger is known straight, reinstall the derailleur, taking care to check that it is threaded properly. Cross threading the derailleur will also cause misalignment.
5. Double check the limit screw positions, with the cable still disconnected. Manually move the derailleur to the top of its range. The upper pulley should be aligned with the center of your largest cog when stopped against the upper limit screw. When at the bottom of the range, the upper pulley should align just slight to the dropout side of the center of your smallest cog.
6. Put the cable back through the barrel adjuster, making sure that the barrel adjuster is positioned to allow it to be turned 2 full rotations in either direction, and pull it across the notched side of the fixing bolt. It matters which side of the fixing bolt the cable lies on. Make sure this is right.
7. Pull the cable tight, and use a 4mm Allen key to tighten it. Torque to 4 - 7Nm.
8. Check your shifting performance. Adjust cable tension as needed until shifting is smooth.

From the sound of the issues you are describing, I expect your derailleur hanger is misaligned.

When troubleshooting rear derailleur setup, these are the basic steps. (I know you said you've checked some of these. I'm still including them for the sake of a complete answer.)

• Disconnect the cable from the cable fixing bolt, and remove it from the derailleur.

• Remove the derailleur from the frame.

• Check the alignment of the derailleur hanger. Many shifting issues start there. Even a new derailleur hanger often needs to be aligned.

• Once your hanger is known straight, reinstall the derailleur, taking care to check that it is threaded properly. Cross threading the derailleur will also cause misalignment.
• Double check the limit screw positions, with the cable still disconnected. Manually move the derailleur to the top of its range. The upper pulley should be aligned with the center of your largest cog when stopped against the upper limit screw. When at the bottom of the range, the upper pulley should align just slight to the dropout side of the center of your smallest cog.
• Put the cable back through the barrel adjuster, making sure that the barrel adjuster is positioned to allow it to be turned 2 full rotations in either direction, and pull it across the notched side of the fixing bolt. It matters which side of the fixing bolt the cable lies on. Make sure this is right.

• Pull the cable tight, and use a 4mm Allen key to tighten it. Torque to 4 - 7Nm.

• Check your shifting performance. Adjust cable tension as needed until shifting is smooth.