Nice to hear from you again!
1) It depends on your home country if verification is necessary or not. If you tell me, I can try to find out more about the procedure in your specific case.
Apart from that: If you have all the required documents and fulfil all the criteria you can get married in Germany.
2) You don’t really need to legalize your marriage in Germany if it’s been performed in Denmark. German authorities usually accept marriages that took place abroad as long as they happened in accordance with the law of said country.
Since your (future) husband is German, he can get a certificate to validate/register your marriage at a German registry office though. But this is not mandatory.
Source (German language): https://www.auswaertiges-amt.de/de/newsroom/buergerservice-des-auswaertigen-amts/faq/04-heirat-ausl---gueltigkeit-in-d/606266
3) You can apply for the German citizenship after 6 years (!) of staying in Germany (the time you spend in the asylum procedure also counts into these years). The law explicitly says that you have to reside in Germany (of course you can travel in the meanwhile), but you have to live here for the majority of the time.
Source (German language): https://www.gesetze-im-internet.de/rustag/__10.html
4) If you are married and get a child abroad: Yes. Since your husband is German, your children will be Germans too.
If you are not married (yet), he has to do the so-called "Vaterschaftsanerkennung" (acknowledgement of paternity) at Jugendamt. He can already do this before the child is born. After this first step your child can get the German citizenship.
Source (German language): http://www.china.diplo.de/contentblob/3485502/Daten/3817836/mbgeburtinchina131216dd.pdf