You need Entomology to get the Spider Egg and it is from Spider Webs, however, it is a very rare drops as Silk is meant to be a 2nd Tier material and the game is not really balanced around having 2nd Tier gear yet. There will likely be easier ways to get the Egg in the future though.
The Bug with the Kids probably has something to do with a re-usable variable being stuck in a parallel process somewhere (something I shouldn't have done but may have by accident). Other gifts will also give you a Colourful Pebble.
The student needs to be either Intimidated, or you can use and Invisibility Spell/Potion and knock him out by hitting him over the head, success chances increase drastically if you have a Wooden Club in your inventory.
Swordsman is supposed to have one of those "long haired handsome looks". He and the Thief are possibly the hardest fights in the game so far. Also, there are no female Travelers, but I do want to add them at some point. I will need to work out some use for them though, like keeping them as slaves in your house or something.
Flax Seed should turn into Linseed Oil, but I think it was an Alchemy recipe. Forest Seeds should turn into Forest Essence, but I think that was an Alchemy recipe as well as it required Flowers too. Normal Oil requires any other seed though.
Botany and Foraging level shouldn't effect the variety of seeds you get as it always pulls results from the same pool of Seeds at every level, it is just a coincidence!
You don't get to choose a specialty at level 2 Cooking, however, you can buy Recipes from the Traveling Merchant. Recipes tend to cover different food items and often overlap in what they teach, so it is possible to get the same recipes as were available at character creation by just buying the region specific recipes from the Merchant. (Though there may be a few left out at the moment).
Glad you like the game!