'cooker-upper' is just a funny, clever way to describe himself as a chef. (as a chef may 'cook up' some food for example) It is not a term that would be used in normal conversation, it is meant to be funny. ^^

I think that adding 'the' to the sentence makes it more clear that you are indicating a particular section of the forest... "Yevtukh brothers were buried in the 19-20 quarter sections of Bykyvnia forested area." or if the 19 and 20 sections are separate, you could say "Yevtukh brothers were buried in the 19th and 20th quarter sections of Bykyvnia forested area." (on a side note, that is fascinating information about the graves!)

Yes, 'semis' is a trading term used to indicate stocks for semiconductor companies.

Yes, you can use 'prove' or 'proof' with claim. It would be the same as saying 'Show me the evidence to back up what you're saying' example: You need to provide proof of your claim. Also: You need to show evidence to prove your claim. Or simply: You need to prove your claim.

The first answer "The fee here is very expensive, because the fee is counted in US dollar" is the most natural sounding, but a native speaker would more likely say 'The fee here is very expensive because it's (counted) in US dollars.' A native speaker would probably drop the word counted. 'The fee here is very expensive because it's in US dollars'

It most likely comes down to simple history. Goiter is thought to come from the latin word 'gutter' which means larynx or bronchus. Before modern medicine defined and classified a goiter as an enlargement of the thyroid, it was often referred to as a bronchocele... probably due to the close proximity of the areas. This article gives a little insight : https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3169859/

Both sentences can mean the same thing, but 'I'm going to do the laundry' can be used in two different ways... 1: I'm going to do the laundry (Present tense: as in, I'm physically leaving to go do laundry right now) - this is the meaning that matches 'I'm off to do the laundry' 2: I'm going to do the laundry (Future tense: as in, I'm not doing it right now, but at some time in the future, I'm going to do the laundry)

'Are you a relative of his?' would be correct from these 2 examples. 'Are you related to him?' would sound less awkward.