1. come forward
a) During our English classes (or lessons?) no one wants to come forward and answer the teacher's questions.--lessons is better.
b) They've come forward with ambitious plans and ideas. (VS "came forward...")---come up with is better here.Choosing simple past or present perfect depends on your context.

2. come into
a) After his grandpa's death, he came into a lot of money.--- ok.

3. come off
a) I really hope that our little party at the end at the end of the weel is gonna come off. (could I use will here? would it make any difference in meaning?)-- I really hope that our little party will come off at the end of the week.If you use going to/gonna it will give the impression that there is a plan.
b) A: What are you doing here? I thought you were (VS "are") in London! B: Nah, it didn't come off. (VS "it haven't hasn't come off")
past simple is okay but I couldn't get the menaing of come off here.
4. come about
a) I have no idea how come such a thing have come about. (could I simply say "came about" ? does the simple past tense change the meaning somehow?)If you use past simple, there will be no relation with present time.In context, it will give you a clue about which one to use.
b) A: My cousin had an accident last week. B: How did it come about?-- ok.

5. turn in
a) "I think I'm gonna turn in as I am very tired". (can I simply say "I think I will turn in..." ?)--yes you can even will is better I think.
b) Let's turn in. (it sounds a bit unnatural to me, but I'm not sure...)-- sounds strange without a context.
c) Let's turn in earlier today. -- ok.
d) We will wait till everybody in the house turn in, and then we will break into it.--ok.

6. patch (things) up
a) We fell out over two weeks ago, and since then we haven't patched things up.--ok.
b) A: I had a quarrel with my wife this morning. B: You had better patch this up. (can I say "I've had..." ?)--I didn't find patch up sensible here.

7. dip into
a) Honey, I think we will have to dip into our car savings in order to tide us over. (any difference between this one and "I think we're gonna have to..." ?) --will have to is better I think.
b) He is so mean. He never dips into his bank acount, no matter what happens (VS "is happening" ?)-- what happens generalise the situation if you use is happening thre should be something going on at that time.I couldn't understand dip into here.

8. dry up
a) My presentation started well but I dried up quickly.---seems okay.

9. cough up
a) Scrooge was such a person that could he/ she hardly coughs up a penny for a poor man.--ok.
b) Mary could hardly cough up a pound or two after loosing her job.--I don't think this is correct.

10. doze off
a) The child burst into tears, but after a short while ithe/she dozed off. (VS "it has dozed off" ?)--simple past is okay.It gives me the sense that it will mean the child still sleeps if you use present perfect.    

Well, I tried to help let's wait for the other posts.