Software for making video lectures for Linux (switching to online teaching with the COVID-19 pandemic)

Question

In preparation for switching to online teaching with the current COVID-19 pandemic, I would like to be able to create video lectures. Unfortunately, I am not aware of many useful tools for this for my OS (I am using Linux, distribution is Arch Linux).

What I would ideally need is:

1. Video, audio capture
2. Screen capture (ideally at the same time)
3. Sketching of graphs, diagrams, etc like on a blackboard
4. Writing equations like on a blackboard
5. Basic video editing (cutting etc.)
6. Ideally, I would like to be able to include one video in the other (e.g., putting the webcam in the lower right corner of the screen capture)

These tasks do not necessarily have to be performed by the same software (although that would be a plus). I do, however, need them to be reasonably stable.

What I found so far:

1. obs-studio seems to be able to do this. Answers to this question recommend obs-studio and various alternatives including ffmpeg, MoviePy, VLC.
2. obs-studio seems to be able to do this. An answer to this question recommends also gtkrecordmydesktop
3. This is problematic. There are a trillion photo/graphics editors (gimp, inkscape, etc etc), plus scientific plotting tools (python-matplotlib etc), but none of them allow sketching a plot in a few seconds. pencil and pencil2d are supposed to do something like this, but pencil does not run on my system. pencil2d is for creating animations, which seems overkill if I am going to do screen capture anyway. Also it does not allow including text, so any drawn text will look really clumsy (really, really clumsy, as in like drawn by an illiterate 4 year old). It could be done with painting applications such as gimp, but they are clearly not designed for this and are full of counter-intuitive features (gimp in particular, with its selections). Inkscape seems possible choosing the freehand calligraphy tool, but even this feels like using the application for something it was not designed for.
4. Similar to 3. It is not difficult to write equations in LaTeX, but you cannot do it while giving a lecture. Inkscape with the textext extension might work.
5. I have heard that obs-studio, blender, audacity can do this, but I have no experience with them.
6. ?

Relevant questions

This question asks about video lecture software for Linux, however, it does not seem to be concerned with a scientific or formal lecture that would need more than video capture.

Edit (Jan 2021):

This is not a completely satisfactory solution, but what I ended up using is:

1. obs-studio
2. obs-studio
3. xournal++ (Not ideal, but can sketch easy graphs and diagrams and schemes in empty pdf pages or in pdf pages with pre-arranged elements, e.g. on the margins of your lecture slides. It can also highlight text, add text, add equations using LaTEX syntax.)
4. xournal++
5. ffmpeg
6. obs-studio as per @Corsaka's suggestion

Answer 1

1. Yes, OBS Studio is your best option here, as it's very customizable.
2. OBS Studio here also.
3. Geogebra is a fantastic free option for the majority of mathmatical things. However, making graphs simply via drawing may prove difficult, and I couldn't find any software that would support this, so you'll probably have to use a simple painting software such as PAINT.NET, Inkscape, or the online Sketch.io.
4. Since you're making a video, you can simply set up all the equations in LaTeX beforehand, then add them to OBS Studio as sources, where you can then move them around.
5. A simple free option is openshot, which is open-source and pretty easy to use.
6. OBS Studio supports this by simply adding a video camera as a source then resizing it.