The small of the back is literally where the back is smallest... well, on a fit person - the waist. Pretty much the opposite of the belly button, a little above where the curvature of the hips meets the spine. The area that covers your lumbar vertebrae.

In Judaism, it is the anniversary of the death of a relative, observed with mourning and the recitation of religious texts. Sometimes a fast as well. (Well, assuming you meant yahrzeit. I'm not aware of a word yahzreit)

Carbon intensive typically means that it has a large carbon footprint relative to its value. Or in other words, the process/product/company or whatever produces a large amount of carbon dioxide. This is a newer term that has arisen with concerns of the environmental impact of products. In this context, it seems to be commenting on the technology used in the creation of whatever exports. The United States' relative lack of concern with the amount of carbon dioxide means that the same product produced there, when compared to another country, would be more harmful to the environment. For example, when comparing modes of transportation, flying a plane is carbon intensive while riding a bike is not. When considering products, more complicated items with higher levels of processing/refining are typically more carbon intensive, while raw materials and basic items are not. But this is saying if you produced a bike in the United States, because of their aging technology and methods, it would leave a larger carbon footprint than the same bike produced in... Europe perhaps.

This is not necessarily saying she did not want it. It's saying that she wants it less than he would like her to. So for example, if he wants to buy a sports car, and he wants her to also want to buy a sports car, she is less enthusiastic about it than he had hoped. So the difference is, if you say "She did not want" then on a scale of 0 - 100, where 0 is not wanting at all, and 100 is wanting dearly, then she is at 0. If you say it as it is originally stated, then all it is saying is that he would like her to be at some number (say... 80...) but she is only at 60. Make any sense?

Clothes for all situations

I like a smart and modern clothes like any other women woman (you are comparing yourself "I", which is singular, to other women. When you use the word "any" as an adjective it refers to a single object. So "any other woman" compares yourself to every other woman individually. To make it extra confusing, this single object could be a group. For example, "Are there any women who like modern clothes?". :)
And I like clothes from a natural textile, for example wool, cotton and flax. It's also possible to say "I like natural textiles, for example...". Not that what you have said is incorrect. I like it when the clothes are multicolor.

I like to wear my light blue dress and the wite white shoes in the summers. I usualy usually take a small handbag with me also. I wear a dark skirt and light shirt , high heel shoes in my work " in my line of work" or "for work" or "to work". This is missing parallel sentence structure. When you make a list of items, use the same form to list them all. For example "I ate apples, bananas, pears, cucumbers, and peaches." or "I ate apples and bananas and pears and cucumbers and peaches". Try not to switch between the two: "I ate apples, bananas and pears, and cucumbers and peaches" as this would imply that "bananas and pears" was in some way a single item. For parties I  usualy usually choose a smart tight dress. I wear the casual clothes for a country walk, for example blue jeans and rose T-short. T-shirt.
But color and moda this is not a word in English, perhaps you mean "fashion" means mean nothing when it's very cold, I wear the thick trousers, gloves, scarf and hat. This last sentence should be reworded or perhaps split into two. It is expressing two different ideas, "Color and fashion mean nothing when it's cold" and "when it's cold, I wear ...". You can join them together using "so" as in "But color and fashion mean nothing when it's very cold, so I wear thick trousers, gloves, a scarf, and a hat." My health is more important to than beauty!
What's kind of clothes do you prefere prefer to wear?

Writing the a letter to the a/my fictitious friend.

Task.
You have received a letter from your English-speaking pen friend (pen friend is correct, in North America so far as I know pen pal or penpal is more commonly used) Bill who writes:
"I don't think it will be a problem for me to choose a job in the future as I'm really interested in foreign languages, culteres cultures and countries and A little bit of a long sentence, as it is expressing two distinct ideas, consider splitting it into two sentences. I hope I'll work as a translator or teacher of foreign languages some day. Have you already decided on your career? What job are you going to choose? Why? What do your parents think about your choice?
I've lived in the USA my whole life but I'd really love to travel to other countries..."

My answer to Bill.
Dear Bill,
Thank you for your letter. I was really pleased to read it. According to your request, I'll tell you about my plans on future! "I'll tell you about my future plans" or "I'll tell you about my plans for the future"
Frankly, I have always been interested in animals! I used to spend hours playing with them when I was a child. Animals often need human's "often need people's help" perhaps? human's is singular, so this is stating that animals need the help of one human. "Animal's often need humans' help" would be referring to the collective help of many humans. Regardless, this is an awkward way to phrase it, and it may be a good idea to consider rewording. help, they are defenseless. I'm anxious to be a veterinarian when I've grown up. This sentence contains a verb tense shift. It's been a while since I've studied English grammar, so you may be more knowledgeable than me, but "I'm anxious" is present continous tense, and "when I've grown up" is ... not? I'm not entirely sure. Anxiety is typically associated with unease. So if what you're intending to express is that you're uneasy or nervous because you are not yet a veterinarian, then I believe the correct sentence would be "I'm anxious to be a veterinarian when I grow up". My parents accept my decision. Actually, as long as I'm happy they don't care what job I do.
As for travelling, I completely share your interest! Which countries are your favourite? The subject and object must agree, countries is plural and favorite is singular. This sentence could be "Which country is your favorite?" or "Which countries are your favorites?"  Who would accompany you? Do you prefer having a guide guided trip or not? Tell me more, please.
I have to go now.
Write soon.
Best wishes,
Olya

First notebook entry: Introducing myself

I live and work in Ottawa, Canada. The city is largely bilingual and there are employment opportunities for those who are fluent in both French and English.

Most of my work expereince has been in half French environments - ski instructing, construction, and the federal government. When I make the effort I can understand a large portion of conversations that I know the subject of, but I'm looking to increase my speaking/writing abilitiy. Really, I'm looking to develop some French language ability without the embarassment of speaking really broken French to fluent friends and co-workers.

I am fluent in English and university educated, and I am more than willing to converse with anyone looking to learn.