Need advice

Using the gay rainbow flag for progressivism is kind of insulting. And you might want space for eco-socialist, as many Greens now identify as that.

Before I go
If I was you I'll pair people based on their political test score, you know the one with the four squares

Don't be so hard on him, he's trying to make an app

I would consider removing Progressivism entirely from the list because it doesn't represent any specific political ideology. Progressivism is about the idea of social progress and moving society forward through policies based on empirical evidence. It can broadly encompass both social democrats and communists.

Very generally, you could break the ideology between these groups -

Anarchists
Marxists
Social Democrats

Between the Anarchists, you could reasonably divide them between Social Anarchists, Individualist Anarchists, and Mutualists.

Between the Marxists, you could reasonably break them down between Orthodox Marxists, Left Communists, and Leninists

Between the Social Democrats you would have, well, social democrats and also democratic socialists.

Here are some good figures for each ideology

Social Anarchism - Mikhail Bakunin
Individualist Anarchism - Max Stirner
Mutualism - Pierre-Joseph Proudhon

Orthodox Marxism - Friedrich Engels
Left Communism - Amadeo Bordiga
Leninism - Vladimir Lenin

Social Democracy - Eduard Bernstein
Democratic Socialism - Salvador Allende

Rename communism to marxism-leninism because 90% of non-leftists who say "communism" are just referring to the soviets and the eastern bloc

there anarcho-communism, trotskyism, left-communism, maoism, communalism

social democracy too I guess

classical liberalism and neoliberalism seem redundant. alt right isn't a political ideology.

It is definitely the two I'd go with if I wanted to choose only two.

They're alright imo

See this is more or less the problem. The left today is not in the mainstream. Since you're not a filthy commie we're probably going to disagree on what "leftism" means. "Democratic Socialism" is probably an alright one to have in there since it exists for some reason, but most other ideologies in your webm are badly named in ways other people have explained. Consider "Social Democracy" (basically scandivanian type shit).

While I won't tell you what software you should make, I would personally remove non-economic ideologies since they're mostly minor things like wanting gay people to be able to marry.

going to bed will probably see response tomorrow

This is not a bad idea, I could include it later on. For the start I want to make it so that you have something approximate position to click on. If I can roughly satisfy Holla Forums, I'll consider my mission complete.

Yes, I understand that many of these positions don't have a coherent ideological backing, but many people identify as them. I am trying to include both what is popular and what has real weight behind it.

I know for progressives, but I legitimately couldn't find a flag. It appears to me that the only differentiating factor between them and others on that list is that they by rule care about sexual minorities. If you provide a better flag I will include it. TYT's logo?

That was along my line of thinking.

Might make an option in the future, like calculating the distance between two positions and pairing people with the highest distance/length.

No, that's liberalism. And TYT is even more insulting, Jimmy Dore is the only principled progressive on that entire shit network.

This is a very difficult question to answer.
I can only answer this one :

3) Flags for these positions which can be downscaled to 29x29 pixels without looking bad
Social-Democracy is often represented by a single rose, with Democratic Socialism (our more radical cousin) represented with the rose and the fist.

4) The antagonism level between various positions of the left you or others in this thread proposed
Socdems are bullied by everyone, from leftcoms to MLs including Trots. Partly because of our history - we did kill Rosa, after all, but also because we technically do not want to remove Capitalism for several reasons - I would say because it's too strong to kill in one blow, our detractors because we are filthy bourgs -, rather improve it and make it liveable for workers.

Wikipedia articles on leftist ideologies give a good synopsis ; they are, however, far from perfect.

I suggest you base your ideological categorisation on the radical-reactionary axis of societal change. This picture is inaccurate as in reality the status quo 'centre' is between conservatives and reactionaries, and the reactionary side of the graph should be as long as the whole progressive line is.
So it would roughly go something like:
Anarchists - Socialists - Social democrats - Social liberals - Conservative liberals - (status quo) - Traditionalists (Paternal autocrats like monarchists) - Fascists - Nazis - Primitivists