With or without to

Както так .

In the room she could see a man sit in an old armchair. (Еще один возможный тут вариант - In the room she could see a man sitting in an old armchair. Первое предложение переводится как - Из комнаты он мог увидеть, как человек сел в старое кресло. Второе предложение - Из комнаты он мог видеть, как человек садится в старое кресло)
We noticed a man clean his shoes. (Такой же вариант - cleaning. Мы заметили, как человек почистил свои туфли. Мы заметили, как человек чистил свои туфли (cleaning - упор на процесс)
The teacher watched the pupils run and play in the garden. (running and playing)
They saw Henry shut the door of the study.
Nobody noticed him come in and sit down.
He felt his heart beat with joy.
I would rather stay at home tonight.
Would you like to go to Paris?

верна ли структура самих предложений

Хороший вопрос)))))))

в Париже - не была.
и ничего сказать об этом не смогу.

а автор вопроса - полиглот,по видимому.

With or without "to":
In the room she could see a man sitting in an old armchair.
We noticed a man cleaning his shoes.
The teacher watched the pupils running and playing in the garden.
They saw Henry shut the door of the study.
Nobody noticed him coming in and sitting down.
He felt his heart beating with joy.
I would rather stay at home tonight.
Would you like to go to Paris?