Are chodzi and ide (I cannot remember if that is the proper 3rd person singular) completely synonymous in Polish?
No. Polish verbs of motion, like in other Slavic languages, distinguishes between determinate and indeterminate verbs.
What that means is that chodzi refers to frequent trips and movement in general. Idzie on the other hand, refers to a a single trip going somewhere specific.
Verbs of motion are one of the bigger points of trouble for native speakers of Germanic and Romance languages, so I'd suggest Googling "Polish verbs of motion" to find out more. This site has a pretty good explanation, but it's gonna take a while to get used to: http://www.polish-dictionary.com/polish-motion-verbs
In different contexts The horse is walking/ The horse goes.
could be correct.
chodzi = has ability to walk / is walking/walks aimlessly/ goes/walks (somewhere) regularly
This particular short sentence seems to me as is walking aimlessly or has ability to walk
It's just different conjugation class. Its infinitive forms are „chodzić” and „iść”.
The voiceover sounded like "Ko łomnie chodzi," not koń. I found it sounded unnatural.
Should the translation not be "a horse is walking" ? The horse is walking I think is "ten koń chodzi" right ?
"Ten koń chodzi" would mean "This horse is walking," as "ten" is a demonstrative, not a simple definite article. There are no articles in most Slavic languages, so no difference between "a horse," "the horse," and simply "horse."
is it like chodzi alone can mean "is walking" or "walks" but if it's with a destination it's only "walks"? and why is spaceruje accepted on some sentences but not on others?
It is an accepted answer, it should have worked.
Remember, though, that normally "chodzić" only translates to Present Simple (to go, to walk). It's only because this sentence is so vague, there's no direction nor destination, that 'is walking' may be accepted. If it was "Koń chodzi do sklepu", it could only be "[The/A] horse [goes/walks] to [a/the] [store/shop]". Not "is walking".
Is the knight chess piece also called "koń" in Polish? Can I say: "Koń chodzi po literze L"?
Colloquially some people call it „koń”, but the proper term is „skoczek” (lit. "jumper"). And you wouldn't say „chodzi” (because it has no legs), but rather „porusza się”. So in short: „Skoczek porusza się po literze L”.
whats the big difference between "Horse walks " and "A horse walks" I didnt get it. Maybe my english is to bad for that lessons :-(
"Horse walks" is not grammatically correct. It requires either "a" or "the" before it. Even though articles don't exist in Polish, you need to add them when you translate into English.