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I would prune off the brown sections, and plan to do further major pruning over the course of 2 more seasons, after that an all over trimming once a year should keep the plant healthy. This article has more information: https://www.gardeningknowhow.com/edible/herbs/lavender/pruning-lavender.htm

Propagating lavender takes patience!
Take your cuttings from the stems that are still green, not the woody stems.
Clean about 2 inches of stem of leaves and dip each stem into rooting hormone.
I like to make a tray of sand and perlite and place each cutting into the damp tray.
Keep damp and out of direct sunlight. I place them in a shady part of my garden.
After 6 weeks, tug the cuttings slightly--if they resist, gently remove them and pot into small starter pots in a good potting mix.
I have had some cuttings take up to 8 weeks to form tiny roots. Growing Lavender this way is tricky for me in my Zone 4, but I have had success and I love the process of making more plants this way.
Hope you enjoy it also!

Here is an article about growing lavender.

https://www.gardeningknowhow.com/edible/herbs/lavender/lavender-in-the-garden-information-and-growing-lavender-tips.htm

Lavender loves hot and dry weather.
Choose a variety that best matches your growing conditions.
I successfully grow Munstead in my Zone 4 garden.
Here are some links for you with more information and available varieties.

https://www.gardeningknowhow.com/edible/herbs/lavender/lavender-in-the-garden-information-and-growing-lavender-tips.htm

https://www.colostate.edu/Dept/CoopExt/4DMG/Plants/lavcont.htm

http://ucanr.edu/sites/scmg/Plant_of_the_Month/July_2007-Lavender/

A few things could be coming into play here. It could be transplant shock, overwatering or perhaps a little bit of both. Lavenders are very sensitive to being overwatered and actually prefer to dry out between waterings. The best way to determine when to water lavender is to touch the soil - if the soil is dry then it is time to water, if it is moist/damp then you should postpone watering.

Here is an article that will help you that discusses how to treat plants suffering from transplant shock after being repotted:
https://www.gardeningknowhow.com/houseplants/hpgen/repotting-stress-treatment.htm

For information on treating overwatered plants, please visit the following link:
https://www.gardeningknowhow.com/plant-problems/environmental/overwatering-container-plants.htm

For information on the care of lavender, go here:
https://www.gardeningknowhow.com/edible/herbs/lavender/lavender-in-the-garden-information-and-growing-lavender-tips.htm

This article lists several insecticide options:
http://www.missouribotanicalgarden.org/gardens-gardening/your-garden/help-for-the-home-gardener/advice-tips-resources/pests-and-problems/insects/hoppers-and-leafhoppers/leafhoppers-and-planthoppers.aspx

For additional information on leafhoppers, please visit the following links:
https://www.gardeningknowhow.com/plant-problems/pests/insects/leafhopper-control.htm

I would do a general search for lavender seed.
There should be many sources that would be able to ship to you.
Here is a link about how to grow lavender.

https://www.gardeningknowhow.com/edible/herbs/lavender/lavender-in-the-garden-information-and-growing-lavender-tips.htm