It certainly appears that Russian influence on politics is negligible in Western Europe - is this incorrect?

It's not as visible in Western Europe, but I think it's incorrect to call it negligible yes -- e.g. I understand that very recently there was some noise in the UK over the Conservative Friends of Russia who used an anti-gay attack on a Labour Party politician who was criticizing Russia's human rights record.

Why is hating the EU necessarily a pro-Moscow position?

It isn't necessarily so. But hating the EU currently helps Moscow policy, so any anti-EU movement will tend to be supported from afar by Moscow, which will in turn tend to lead such movements to end up holding pro-Moscow positions.

e.g. The russian textbook Foundations of Geopolitics says "United Kingdom should be cut off from Europe."

Please explain how the political parties map onto your axis in, say, Britain.

Where I use 'universalism' to mean 'focus on equality, democracy, human rights'.

Liberal-Democrats -- very Universalist, very pro-EU, significantly anti-Russia
Labour Party - somewhat Universalist, somewhat pro-EU, somewhat anti-Russia (currently -- it used to be different)
Conservative Party - minimally Universalist, recently getting more anti-EU, and recently getting cozier with Russia
BNP - very anti-Universalist, extremely anti-EU, extremely pro-Russia.

So I think that the pattern I observe holds even in the UK. e.g. The Conservatives are the main anti-EU party in the UK, and they're also in the same parliamentary group in the Council of Europe as Putin's party...