Hi! I have a half-dying plant that I would like to try to revive and I thought knowing more about it would help that goal.
I will add pictures and try to describe it. I am not experienced in taking care of plants, but I want to try and learn. I live in Sweden.

Description:
It branches off in pairs, with the branches in a pair on opposite sides of each other. The next pair is 90 degrees turned around, so that the third pair is directly above the first. Making a cross shape from above. The leaves follow the same pattern.

Every centimeter or so (more often further from the base) there seems to be a tiny initiation of a branch pair growth, making small ridges along the branches. Towards the end these are where the leaf pairs grow out from.
Check the pictures for a closeup on the leaves.

Along the branches there are bush-like outgrowths from the ridge-pairs. I'll describe them as hairs. (In the picture, some of these are brown, but they are stiff and fall off if I touch them. The live hairs are white.)

The revival seems to have hope. A single flower has shown at the top of one branch. The petals on the flower follow the same pair system as the rest of the plant. Their shape is round ending in a point.

Fungal growths:
Along the sides of the soil there are white fungal/mucus (I'm no expert in the difference) as show in the picture. Is it harmless? Should I try to remove it?

Pictures: (if the attachments didn't work properly)
Identifying my flower - Imgur
Imgur
Imgur
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Trying to identify this half dead flower. - Imgur

I'm not a plant, pest, or fungus expert, but have grown plenty of plants.
To me it looks like:
A plant that struggled to get through winter without enough sun.
Potting soil with lightweight inert white foam added, commonly called perlite in the US (but only you can tell us what it feels like to the touch).
A plant that has over grown it's pot size.

But if that's fungus, I would throw out the entire plant.