Birinin elleriyle yapabildiğini, ellerimle yapsaydım, onları dinlendirirdim.

This is my version

If someone was so kind and helped me with these.

1. Wait a second before you begin speaking. Konuşmaya başlamadan önce biraz bekle.

2. The children waited patiently for the show to begin. Çocuklar temaşanın başlamasını sabırla bekledi.

3. The musicians began by tuning their instruments. Müzisyenler çalgılarını akort ederek başladı.

4. Once you have begun the second section of the test you cannot go back to the first. İkinci parçaya başladıktan sonra birinciye geri dönemezsiniz.

5. They have begun a textile business in Australia. Avustralya’da bir dokumacılık işine başladılar.

6. Are they finished with repaving the streets? Sokakları yine döşeme işiyle bitirdiler mi?

7. When they finished laughing they were on their way to become not just friends but very good friends. Gülmeyi bittikleri an sadece arkadaş değil iyi arkadaş olmaktaydılar.

8. Finland finished the campaign with a dreadful total of ten points from ten matches. Finlandiya mücadeleyi on maçtan aldığı on noktanın iğrenç toplamıyla bitirdi.

9. What would happen if we stopped vaccinations? Aşılamayı bıraksak ne olurdu?

10. The browser has stopped restoring this website. Tarayıcı bu siteyi eski durumuna getirmez oldu.

11. Athens denies it will have to quit eurozone. Atina euro bölümünü bırakmasının gerektiğini inkar ediyor.

12. If possible, you had better quit milk. Mümkünse sütü bırakırsan iyi olur.

13. What might be the problem if my PC continues to reboot? Bilgisayarım yeniden başlayıp duruyorsa sorun ne olabilir?

5. After driving for 40 years I slept in the wheel and ended up in an accident.

Kırk yıl sürdükten sonra direksiyonda uyuyakalıp kazaya uğradım karıştım.

The translation here suggests the driving that preceeded the accident had lasted full 40 years (a clear entry into GUİNESS BOOK OF RECORDS). Unless this is what is really meant, a verbatim translation should be avoided here and a sentence that would make better sense in Turkish must be used.

KIRK YILLIK  ARABA SURME TECRÜBEME RAĞMEN, DİREKSİYONDA UYUDUM VE BİR KAZAYA KARIŞTIM.

KIRK YILLIK  ŞOFÖRLÜK TECRÜBEME RAĞMEN DİREKSİYONDA UYUYUP [UYUYUNCA]  KAZA YAPTIM.

This way is more natural in Turkish. As "karışmak" can mean " to involve in something that was caused by other people " but here the person himself  caused the accident by sleeping on the wheel.

It was really ment  -  or maybe originally not  -  but this is how crash reports for insurance companies sometimes look like.

                          http://msoe.us/taylor/humor/accident

Enjoy. I like them a lot.

Sağ ol, gokuyum.

A little help here, please.

1. You’ll be leaving for Tokyo soon, won’t you? Kısa süre içinde Tokyo’ya gitmek üzere yola çıkıyorsun, değil mi?

2. I really think you had better see a doctor. Bence gerçekten bir doktora görünsen iyi olur.

3. Fiestas have been made in Cologne, Germany for many years now. Fiesta’ları uzun yıllardır Almanya’daki Köln’de yapılıyor.

4. She wished she had bought a new car. Yeni bir araba satın almış olmasını diledi.

5. Which model goes 250 k.p.h.? Hangi model saatte 250 km hızla gider?

6. In his position as managing director, he is responsible for more than 300 employees. Onun makamında müdür olarak 300’den daha fazla çalışandan sorumludur.

7. Ken said he would finish that project soon. Ken o projeyi yakında bitireceğini söyledi.

8. Unfortunately, Mary has few friends in Tacoma. Yazık ki Mary’nin Tacoma’da az arkadaş var.

9. Where was Jack yesterday?  -  I don’t know. He might have been seeing the doctor. Jack dün neredeydi?  -  Bilmiyorum. Doktora gitmiş olabilirdi.

10. Jack told her that he was going to come the next day. Jack ona ertesi gün geleceğini söyledi.

11. He drove the car out of the garage and left for work. Arabayı garajdan sürüp işe gitmek için ayrıldı.

12. Where do you think Anne was yesterday?  -  She must have been at home. Anne’nin dün nerede olduğunu düşünüyorsun?  -  Evde olmalıydı.

13. He has no interest in continuing the project. Projeyi devam etmekle ilgilenmiyor.

14. Unless he comes, we won’t have much to talk about. Gelmezse konuşacak çok şeyimiz yok.

15. Look at those clouds! It looks like it’s going to rain. O bulutlara bak!  -  Yağmur yağacak gibi görünüyor.

16. I could hardly make out the ship in the distance. Uzaktaki gemiyi zorla farkettim.

17. That is the man whose grandfather founded the company. Büyükbabası şirketi kurduğu erkek budur.

Hi, I always think that these examples are up to the stories. Because sometimes we use a "tense" that fits on the story but if there is no story, "tense" sounds unnecessary or weird.

As there are no stories,  I will put just some points but it doesn´t mean only my points are correct.

3. Fiesta´ları

Do you mean someone´s Fiestas? Maybe just like

"Onların Fiestaları çok sağlam" Fiestas of those are very sturdy.

Then Onların Fiestaları...

Otherwise;

3. Fiestas have been made in Cologne, Germany for many years now. 

 Fiestalar uzun yıllardır Almanya’daki Köln’de yapılıyor.

***

4. Both "She"s are same she? Then;

4. She wished she had bought a new car. 

 Yeni bir araba satın almış olmayı dilerdi.

***

6. Bit difficult to explain for me but this sounds better for my ears;

6. In his position as managing director, he is responsible for more than 300 employees. 

Bir müdür olarak o makamda 300’den fazla çalışandan sorumlu.

or

Onun pozisyonunda bir müdür olarak 300’den fazla çalışandan sorumlu.

***

8. arkadaş+ı var

8. Unfortunately, Mary has few friends in Tacoma. 

 Ne yazık ki Mary’nin Tacoma’da az arkadaşı var

***

9. olabilir; if you know that he didn´t see doctor, then you can say "olabilirdi". But as you still don´t know, you should say "olabilir"

-I don’t know. He might have been seeing the doctor

 -Bilmiyorum, doktora gitmiş olabilir.

***

11. çıkarmak sounds better for me but still it sounds weird because I don´t understand why would you emphesize that he get the car out of the garage?

11. He drove the car out of the garage and left for work. 

 Arabayı garajdan (dışarı çıkarıp işe gitmek için ayrıldı.

***

13. Proje+ye devam etmek;

13. He has no interest in continuing the project. Projeye devam etmekle ilgilenmiyor.

***

"Şu" is more effective for "fingering" things (just like showing things with your finger);

15. Look at those clouds! It looks like it’s going to rain. 

 Şu bulutlara bak!  -  Yağmur yağacak gibi görünüyor.

***

16. I could hardly make out the ship in the distance. Uzaktaki gemiyi zorla farkettim.

Good but maybe better; Uzaktaki gemiyi (çok) zor farkettim.

***

That sounds pretty weird for practical usage but;

Bu, büyükbabası şirketi kuran adam.

17. That is the man whose grandfather founded the company.

On the other hand, some sort of correct words may be;

Büyükbabası şirketi kuran adam bu (adam).

A big thanks for the corrections and explanations, caliptrix. At this point I always wish I had done a little bit better.

One question:

It wouldn´t have come to my mind to use aorist here. I don´t even know how to ask about it...

Maybe it is easier to come closer to it without the past tense. Yes, I have noticed that dilerim is used as often than diliyorum. It is very difficult to understand the difference. It is the same as with istemek. If someone says

                        Seninle konuşmak isterim

what does she mean by not using istiyorum? Is it about time (yes, she wants to talk but maybe not now), wanting less (actually she doesn´t want but she is being polite) or wanting more (this wanting is a part of her which is not tied to a certain time)?

4. She wished she had bought a new car. 

 Yeni bir araba satın almış olmayı dilerdi.

If you say "diledi", I can´t make it have sense for me. The problem is there is another action, "to buy" and it has past tense as well. If this were a simple verb, I would understand;

Yeni bir araba satın almayı diledi.

That means she wished to buy a new car.

But now we say that she wished in past; and she wished something for in past. Which makes everything complex, difficult to explain.

As she wished, she "did" this in past, and she wished something in past so it is fictious, not real. We must say it so; dilerdi / isterdi;

... yapmış olmayı isterdi.

... gitmiş olmayı dilerdi.

***

The other point you wrote is dilerim and diliyorum; I remember I explained this somewhere in this website, but it is hard to find it or write it completely from some years ago. So I will try to remember and write some shortly;

Firstly you explained right, it is about time.

Secondly it is about emphasizing;

when you say diliyorum, istiyorum, teşekkür ediyorum; I think they have more power, sometimes like exagrating or putting your emotions more (anger can be as well)

İyi bir gün geçirmenizi diliyorum. (feelings can be; now are we done? I have work to do and I want to leave now)

Bir ara sizinle görüşmek istiyorum. (feelings can be; please answer quick, when are you ok for meeting?)

Yaptıklarınızdan dolayı teşekkür ediyorum. (feelings can be; I know you want a thank, so I am giving that to you but I am not that thankful)

or the feelings may be so different from what I wrote here.

But if you want some more "neutral" or "emotionless"/professional approach; aorist is always with you;

İyi bir gün geçirmenizi dilerim.

Bir ara sizinle görüşmek isterim.

Yaptıklarınızdan dolayı teşekkür ederim.