Welcome to Wefugees and thank you for sharing your situation with us here!
First of all: Yes, you can apply for asylum in Germany.
If you apply for a Schengen tourist visa for Germany, you can stay in the country for the period of time your visa is valid for. Then you can apply for asylum when your visa ends, on the very next day (of course also while your visa is still valid). During your asylum procedure, you have an "Aufenthaltsgestattung" (temporary permission to stay).
Here is a detailed description of the steps you need to take by the "BAMF", which is the Federal Office for Migration and Refugees in Germany (they will be responsible for your asylum claim): http://www.bamf.de/EN/Fluechtlingsschutz/AblaufAsylv/ablauf-des-asylverfahrens-node.html
If you want to apply for a Schengen visa, this seems to be the right website to check - TLScontact/ the German Visa Application Centre in Egypt: https://de.tlscontact.com/eg/cai/index.php
Here is further information on all the documents you will need to hand in: https://www.germany-visa.org/germany-visa-requirements/
Some people also apply for asylum while they are on transit - a different procedure applies here, the "airport procedure": http://www.bamf.de/EN/Fluechtlingsschutz/Sonderverfahren/FlughafenVerfahren/flughafenverfahren-node.html
Please note that not every airport in Germany has the necessary facilities.
There is another quite helpful article by Deutsche Welle on "How to apply for asylum in Germany": https://www.dw.com/en/how-to-apply-for-asylum-in-germany/a-18723501
In case you are interested in reading through a few cases we have had in the past, please have a look at the questions and answers from our platform here. They can be very helpful in understanding what you would need to be aware of:
What are the steps to take in order to apply for asylum in Germany?
Can a tourist visa holder apply for asylum?
In the International transit zone in Frankfurt airport - who should I ask for help to start my asylum application?
I hope that these sources help you to get an overview of how the asylum procedure in Germany works and what you need to do - I understand that it can be very complicated! Of course you can get back to us if you need any further assistance.
All the best and take care,