One general point about the EU: for reasons of non-discrimination of same-sex partnerships, whenever I say "marriage" in the following paragraphs, Germany (and probably every other EU country) includes partnerships that are not an official marriage, because their country of origin does not allow same-sex marriages. So when I say there is a family reunion visa for married couples, it does exist for couples that are not married, but can provide proof of their long term relationship. I don't know what would be enough as proof, probably shared bills, shared living, something like that. But since you said you are hiding your relationship, I will assume that you are spending a lot of energy to make sure that no such proof exists. So this is not the route for you. You will need a marriage certificate.
How can we get married in Germany and ask for a family reunion or something similar so that we can live together?
You can get married anywhere you want (including Germany) and then ask for a family reunion visa.
Be warned, the whole process is covered in good old German buerocracy. It will feel like they don't want you to come here or marry. Germany is neither racist nor homophobic in that regard, I can attest that it feels the very same way even as a straight white male German citizen. They are just mean to everyone equally. But it is frustratring and exhausting.
What is exactly the process and its required documents?
You obviously need identification papers. I assume you are coming by plane using an Egyptian passport, that is already covered.
You need an appointment with the Standesamt (civil administration office in charge of marriages) and even if you have all papers perfectly lined up, best case is they will give you another apointment in a few weeks for the actual wedding ceremony. Marriage is no walk-in option here. Be preprared to wait a lot.
As far as papers go, you need to be 18 or older, you need your id (passport), your birth certificate, and a document certifying that you are allowed to marry. This comes from your government and I don't know what it looks like for Egypt. It does not mention who you are eligable to marry, it basically just says you are not already married to someone else. All those papers need to be translated and apostilled by certified persons/authorities. You must not be blood related, and I think the middle east has a weird rule that you may not have been breastfed by the same women as an infant. Germany asks about that as a courtesy to that region/religion, but there is no proof neccessary.
This is a lot of paperwork, that can drag along months and cost thousands of Euros for certified appostiles and translations alone. Since you probably cannot get those documents from Egypt while physically being in Germany, it might also arouse suspicion. None of those documents actually say you are gonna marry or who, so it might just look like you are getting ready to marry a nice German girl, but it is a lot of paperwork and it makes it more likely that someone in your family will ask questions.
The only thing that is not a problem when marriage is concerned, is your visa. You can come here on a tourist visa and marry. Nobody asked us, I don't know if they maybe checked the passport and made sure that the person is in Germany legally, but a tourist visa (or really any other kind of legal stay) is enough. The length of stay of a tourist visa might not be enough to go through the whole process of appointments and papers, but legally, any visa is fine.
Once you are married you need to apply for a family reunion visa. You need to be married (in Germany or elsewhere, does not matter), have a grasp of the German language of level A1 (hint: they will quiz you, there is no point in buying a fake certificate, you really need to do this) and your partner needs to prove they can support the both of you. Assuming he has a steady job and rents a place that is large enough for the both of you, that is perfectly fine as proof.
Here is the problem: You can only apply for a family reunion from your home country. And doing so will require you to disclose the fact that you are married to a man in Germany. Whether your family will ever know, I cannot say. I have no idea how your government works or how leaky it is concerning private information.
There exists a visa that is explicitely created to make this process more streamlined. It's called a "Heiratsvisum" (marriage visa). But while it's more streamlined in the way that you can collect all the papers and qualifications needed to marry and apply for the family reunion and then only travel to Germany once, marry and then stay, it also means you need to collect all the papers and apply in the embassy of your home country before you are even married. Same warning applies: I have no idea how leaky that process is in your home country.
As Germany accepts any marriage, you may want to go look for countries you both can travel to and that accept walk-in marriages with less paperwork required. In the US for example it takes your passports and a 48h waiting period to get married. The flight is probably cheaper than even just the translations of the paperwork you need for German authorities. Plus you need no paperwork from your local authorities. Or look into the Netherlands or Denmark, who legalized same-sex marriages long before Germany and have some experience with that kind of "tourism" even from Germany itself.
However, the "problem" remains that as a citizen of Egypt, no exceptions apply to you and you have to go back to Egypt and apply for family reunion from there, even if you are already in Germany legally, for example on a tourist visa during your wedding.
If I were you, I would start with a German language course right now. Because that is not in any way connected to your sexual preferences and pretty innocent. And you will need it, no matter what and when you need it, it cannot really be expedited. The easiest way might be for your partner to arrive in Germany, settle down, get used to their job and new conditions and then go to the Standesamt and get a list of what is needed. Because in the end, no matter what the internet (or even official webpages) say, that person has to sign off on it, so you have to do what they say and bring what they want.
Another option would be to get married before your partner moves to Germany. That would need to happen in a different country though, there is no way you manage to work through the German paperwork and appointment chains for both of you on a short tourism visa. The visa your partner is using for work will very likely include the option to bring their married partner (and probably even kids, though that does not apply in your case) or apply for a family reunion visa at the same time so you can arrive together.
Whichever way you go, prepare for paperwork and seemingly pointless rules. That is Germany.