Do Delta Pop antennas require a ground plane?

Yes, the Delta Pop Antennas are a1/4 wave length design and require a ground plane.  

What is a grounded element VHF antenna?

Both Delta Pop VHF Com antennas are grounded element, they will show a DC short when measuring from the center pin of the BNC connector to the shield with a Volt Ohm Meter (VOM)  and approximately 50 ohms impedance at the design frequency. This can be confusing when checking coax cables for continuity as it will appear the coax is shorted if connected to the antenna.

Can the antenna be painted?

The antennas can be painted to match the aircraft scheme with non metallic paint with no RF performance loss.

What coax should I use?

Although RG58 can be used, I suggest the higher quality RG400.

Should I use a gasket under the antenna base?

I suggest mounting the antennas per the EAA video linked here: https://www.eaa.org/videos/hints-for-homebuilders/1136841468001

In addition to the info in the video I suggest masking the antenna base and the fuselage so the sealant forms a neat 1/8” wide bead when the tape is removed.

When mounting the antenna, do I need to remove the fuselage paint under the base?

No need to remove the paint as the antennas make electrical contact to the ground plane thru the mounting studs. Many use “star" type washers under the nuts to improve the contact.