1. Is there a module that provides gender selection?
You could modify the Profile module for that.
But how would you like to use it?
2. Birthday module?
Yes,https://github.com/XoopsModules25x/birthday
3. Will it be possible to set every gender their own default avatar?
If you would like to do it as an Admin, you could modify the Profile module for it.
4. Sometimes, when I clean the cache, it remains. Is that ok?
What do you mean by "it remains"? Which cache folder? What files? Normally it should clear the cache in /xoops_data/caches/ folders
5. My website looks messed up on different screen resolutions. How to fix that?
The Default theme is not responsive, so it might not look good in certain resolutions. You need to use a "responsive" theme, like the xswitch and xbootstrap that are provided with XOOPS
6. Is there a way to show new private messages in a custom block (with info and a pic of the sender)? The user will be able to press the button to answer or close the message.
I believe, you should be able to create such a custom block for it.
7. Is it possible to show if the user is online? Example: "Online" text under their profile picture.
We have a block "Who's Online" that show current online users. I assume that you could add such option to check in Profile if the user is online, but it would be hack, and you would have to recreate it in the next XOOPS version.
Let us know if you have any other questions.