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там, здесь relates to the static position of the subject while туда, сюда describes the destination of motion or transfer.

я сажаю дерево там - I am there and I am planting the tree.
я сажаю...

That's what I mean. Examine teeth = смотреть зубы (like a dentist). But to examine teeth one should look into a mouth, смотреть в рот. Here we have a metonymy when we spread the meaning of рот to...

Let's try.

Смотреть + Accusative (without prep.)
a) the same as рассматривать, осматривать, оценивать - to inspect, to examine.
b) to watch the performance, the action (представление, шоу,...

You can check morphology of Russian words here. Enter word "погрузиться" (or whatever) and press "View".

Я бегу за ней.

<<я бегу после неё.>> means actually "I am going to make my run after she will make hers."

Usually translation of separate words does not work without the context.

Right now I can remember phrases like "In enduring grow strong.", "Endure my sword, bastard!", "we must endure to the end" or something like that - I am not sure if they are exact quotes.

what...

In my experience English word "to endure" in most actual situations can not be translated directly as one verb. Mostly one need to rebuild the sentence in translation to reproduce the sense.

Yes, вносить/внести - to carry smth. into smth.; вводить/ввести - to lead smb. into smth. Also it can mean to introduce (to add smth. to smth.), to inject and some other meanings. They are different...

It is correct, but
1)A top of a mountain is "вершина" rather than "верх".
2)Preposition "с" here requires Genitive.
3)Native speaker would say "спускаюсь" instead of "схожу".