Please, check it in English
1.The doorbell rang while I was eating with my family.
2. The car began to make a strange noise when we were driving.
3. Jessica fell asleep while she was watching TV show.
4. The television was on, but nobody saw it.
5. I saw Sue in town yesterday, but she didn't see me. She was looking the other way.
6. I met Tom and Jane at the airport a few weeks ago. They were going to Paris and I was going to Rome. We had chat while we were waiting for out flights.
7. I was cycling home yesterday when a man stepped out into the road in front of me. I was going quite fast, but luckily I managed to stop in time and didn't hit him.
8. How fast was you driving when the accident happened.
9. I haven't seen Alan for ages. When I last saw him, he was trying to find a job.
10. I was walking along the street when suddenly I head footsteps behind me. Somebody was following me. I was scared and I was starting to run.

Question

Please, check it in English
1.The doorbell rang while I was eating with my family. 
2. The car began to make a strange noise when we were driving. 
3. Jessica fell asleep while she was watching TV show. 
4. The television was on, but nobody saw it. 
5. I saw Sue in town yesterday, but she didn't see me. She was looking the other way. 
6. I met Tom and Jane at the airport a few weeks ago. They were going to Paris and I was going to Rome. We had chat while we were waiting for out flights. 
7. I was cycling home yesterday when a man stepped out into the road in front of me. I was going quite fast, but luckily I managed to stop in time and didn't hit him. 
8. How fast was you driving when the accident happened. 
9. I haven't seen Alan for ages. When I last saw him, he was trying to find a job. 
10. I was walking along the street when suddenly I head footsteps behind me. Somebody was following me. I was scared and I was starting to run.

Answer

Hello Natalya. These seem very good. Some of these sentences include complex constructions but most of the time they are perfect.

3. You need an article before TV show. If in a previous sentence you had mentioned the TV show use "the" if it hasn't been introduced use "a."

4. This sentence does make grammatical sense but "The television was on, but nobody was watching it" would be better.

6. Another article needed "We had a chat..."

8. Use "were" instead of "was." "How fast were you driving..."

10. I was scared and I started to run.

Answer

Super good job, Natalya!  Just a few small corrections:

3.  Jessica fell asleep while she was watching THE TV show.
3.  The television was on, but no one was watching it.
6.  We CHATTED while we were waiting for OUR flights.
8.  How fast WERE you driving when the accident happened?

Great job!!  I hope this helps you.

Answer

number 4- If by using (saw) you mean nobody noticed that the TV is on , that is correct. otherwise, it's better that you use (watch) and in this case, I suggest past continuous, was watching ... 
number 6- If you want to use past perfect (had+ past participle) so had chatted.
 If not, you can use chat as a noun but make sure to an article before ( a chat ) 
number 8- Were not was
Good luck :)