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This could be Swallowtail Butterfly caterpillars. They commonly are found on Dill, though they love Parsley too.

Washing and rinsing your Dill should be all you need to dislodge the caterpillars.

Any seeds obtained from a spice jar may or may not grow into plants. Most will have poor germination, because seeds sold as spice are not stored in the proper way for maintaining the health and viability of the seeds. You could try germinating some in a paper towel, but it's more reliable to buy garden dill seed.

This article explains how to grow it from seed:

https://www.gardeningknowhow.com/edible/herbs/dill/tips-on-how-to-grow-dill-weed-plants.htm

Yes, there are many different varieties of Dill.
It may not be easy to identify culinary varieties unless they are labeled.

Here is a great article the highlights some of varieties that are commonly grown.

https://www.gardeningknowhow.com/edible/herbs/dill/dill-plant-types.htm

You can move your plants outdoors but you will need to do this slowly.
It is called hardening them off.

You need to gradually get them used to the conditions outdoors.
Here are some links with more information.

https://www.gardeningknowhow.com/garden-how-to/propagation/seeds/how-to-harden-off-your-seedlings.htm
https://www.gardeningknowhow.com/edible/herbs/dill/tips-on-how-to-grow-dill-weed-plants.htm

This article will help you.
https://www.gardeningknowhow.com/plant-problems/pests/animals/managing-lizard-populations.htm

Dill is technically a biennial. This means that it will grow, vegetatively, the first year, and seed the next. However the seeds typically will take its place the next season.

Here is a link for more information: https://www.gardeningknowhow.com/edible/herbs/dill/