Honeysuckle vines do not need to be deadheaded.  They will drop old flowers naturally.  Here is the information I found on propagating:

Stem Cuttings

Honeysuckle stem cuttings taken near the tip of a growing stem during the summer provide the best cuttings. The ideal stem cutting has two leaves near the tip and measures roughly 6 to 8 inches long. Stems with obvious signs of damage from insects, disease or malnutrition are less likely to take root. Taking two to three stem cuttings for each new plant you intend to propagate helps ensure a successful project and allows you to pick the strongest plants from the cuttings that take root.

Preparing Stem Cuttings

The best growing medium for honeysuckle stem cuttings is a fertile soil that drains quickly in a plant pot with several drainage holes. You can combine peat moss, perlite and coarse sand in equal parts to create an effective rooting medium. Treating the base of your stem cuttings with rooting hormone improves their ability to form new roots. The best way to apply rooting hormone is to make a pair of 1-inch-long cuts in the stem near its base before immersing the bottom half of the stem in powdered rooting hormone. The best time to apply rooting hormone is right after you take the stem cutting. After you've treated the stem cutting, it is ready to plant. Planting your stem cutting quickly after it is taken from its parent plant increases its chances for success.

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Stem Cutting Care

Stem cuttings require little water until they begin to produce roots. Water the soil around your stem cutting often enough to keep the soil slightly moist. Growing mediums that are saturated with water can cause your stem cutting to rot, preventing it from taking root. Until your stem cutting has produced a new root system, it cannot absorb water. Moving your stem cutting to an area with indirect sun and covering it with a plastic bag help reduce water loss. In most cases, your honeysuckle vine should produce new roots within two to three weeks.

Serpentine Layering

Serpentine layering is a simple technique that allows you to propagate honeysuckle, as well as other vining plants where the parent plant is growing. This technique uses a long healthy and flexible stem on your honeysuckle vine that is growing near the ground. Weave the stem so you bury every other bud into the ground, burying each bud approximately 3 to 4 inches deep into the soil and leaving the tip of the vine exposed. Leave the alternate bud next to buried one unburied. Each buried bud will develop roots and produce a new plant. Keep the planting site moist by watering regularly, saturating the soil to the depth of the buried bud sections. This process works best throughout the growing season. Carefully dig around the buried sections after four to six weeks to check for root development. Once the buried sections of your honeysuckle vine have rooted, you can sever the portions attached to the parent vine and transplant into containers or directly into the ground.