If there's old English, are there older versions of other languages like Spanish, German, Russian, Chinese etc?
The simple answer is yes, on all counts. All languages change. They're changing right now. The English (or whatever you speak) is not the same as the English (whatever) of your parents. You might not be aware of the differences, but they're there, and they're small, but they build up over time.
English developed from an earlier stage that we call Middle English. Actually English and Middle English are not set things. These are just arbitrary labels. We call the English after 1066 Middle English, but again since language is constantly in a state of change, there was no one single Middle English. It's just a useful label for us to talk about these.
Old English was the English before 1066. However the further back you go, the more like the other Germanic languages it would have sounded, until eventually it's just Proto-Germanic.
Modern Spanish can be brought back to Old Spanish, what was spoken in the first half of the last millennium. Go back further and it's a variety of colloquial Latin, as was French, Romanian and the rest of the Romance languages. "Romance" here doesn't mean flowers and a candle-lit dinner, but instead means tied to the Romans.
Modern Mandarin is one of a dozen Chinese languages. Cantonese, Hakka, Wu are three others. These developed from what's called Middle Chinese, which was the parent form of the language as spoken around the 7th century and just before. Going back much earlier, you get Old Chinese. Middle Chinese was one path of development that happened from Old Chinese. There were other branches too, but they are now extinct. One such language is Bāshǔ which was originally spoken in the Sichuan Basin but is now long gone, and has been for two millennia.
Did they sound very different as their modern counterparts?
Absolutely. Brasilian Portuguese sounds quite different than Latin would have. A speaker of Modern Mandarin wouldn't be able to easily understand Middle Chinese, and a speaker of Middle Chinese would be utterly helpless at trying to make sense of Old Chinese. Same goes for the rest. If we look at a period of a thousand years, you can pretty much guarantee that most spoken languages will be completely unintelligible to modern speakers.
There are exceptions, such as cases where a community has intentionally changed their language to conform to an older standard. This happens, as people have misguided ideas of language "purity" (which doesn't actually exist as a thing), so you'll often find people throughout history arguing to take their language "back to a purer form", but again, it's all arbitrary.
And have some languages barely changed or even not at all?
There are no natural spoken languages that have barely changed at all. I touched on this above, but I'll reiterate: Any natural spoken language is going to be in a constant state of change, and only through enforcement of older arbitrary standards will there be some slow-down, except often not really because then what you get is a spoken form and a written form which differ, and which people will identify as the same thing, but which actually are not.
Let me know if you want expansion on any of the above points. I just woke up so this this is a little brief.
Also, if you're interested in this sort of thing, I highly recommend taking a look at an introductory Historical Linguistics text, because that's exactly the sort of thing the subspecialty is about. Any of the following would be a good place to start:
tl;dr: Yes. It's all changing all the time.
I do not know much about European languages, but I can tell you about the Arabic language.
Since the introduction of the Quran, the Written Arabic Language did not see much change since it gave people a guideline to follow for the language. The Fus-ha, as it is called, remains the same as it was a thousand years ago. The difference between classical Arabic and Modern Arabic comes when talking about the spoken language. There is no set dialect people learn in school, except for the fus-ha which is just standard Arabic.
Spoken language is so different in modern Arabic that a speaker in Morocco is almost unintelligible to an Arabic speaker in Syria or Lebanon. In many Arab countries that were colonized by Europeans tend to use French or English phrases in their speech, for example in Syria we use Merci for thank you and Bonjour for hello instead of Shukran and Marhaba.
I can shed some light on the development of the Czech language, which I am sure will be very helpful! A Czech speaker would definitely understand a compatriot from roughly two hundred years ago rather easily, and could – albeit with some difficulties or with the help of a transcribed text – understand someone from over twice that long ago, give or take a century.
Proto-Czech and Early Old Czech spanned a period from about the 10th century to the 13th century and both would be extremely difficult bordering on outright impossible for a present day speaker to understand. Some of the words are similar or familiar in the same way one might recognize a word or two in another Slavic language, however even these would require a transcription in the first place, as both the spelling and the pronunciation were vastly different.
Medieval Czech starts to become less illegible, especially with later texts, but spelling and pronuniciation would pose a major challenge. The orthography reform from the early 1400s attributed to Jan Hus would prove helpful to a 21st century person – before the introduction of diacritics the language might remind the reader of Polish with its use of digraphs ("cz") instead of diacritics ("č"). Certain transcribed texts could be highly compatible with today's language, compare "Dřěvo sě listem odievá, slavíček v keřku spievá" which would be rendered as "Dřevo se listem odívá, slavíček v keříku zpívá" today.
The following centuries would see a present day reader gradually achieve an ever greater understanding of the language, although Habsburg rule and the Germanization policies did make the development of Czech more difficult. Spelling would still be noticeably different from modern Czech, but for example the first complete local translation of the Bible, the Kralice Bible from the late 16th century, is highly comprehensible with a modern transcription (and not unreadable without it either), with lexical differences being perhaps the greatest obstacle.
Compare Revelations 1:8, "Jáť jsem Alfa i Oméga, totiž počátek i konec, praví Pán, kterýž jest, a kterýž byl, a kterýž přijíti má, ten všemohoucí," from the 16th century translation (but a modern transcription) with its 20th century edition, "Já jsem Alfa i Omega, počátek i konec, praví Pán, který je a který byl a který přichází, ten všemohoucí."
For a perfectly understandable form of Czech without any major obstacles including orthographic ones to someone from our time, look to the culmination of language development during the Czech National Revival in the 18th and 19th centuries. Josef Dobrovský and Josef Jungmann and their works on Czech grammar and vocabulary formed the basis of the Czech language as we know it today, although the structure of the texts can obviously be sometimes rather archaic.
German
I'd like to single out German for this answer as the answer for German is a little more complicated than for the others. What we know today as Standard German (Hochdeutsch) was historically only a written language which evolved from a kind of artificial dialect of German used by Martin Luther in his translation of the Bible and was based on various dialects of German spoken around Saxony at the time, and has only recently become used as an actual spoken language (and in some parts of Germany, particularly in the South, many people still prefer to speak in their local dialect and have a surprisingly poor command of the standard language). The closest thing there is to 'old German' is what's called Old High German, which is the group of German dialects that was ancestral to the dialects spoken in Central Southern Germany, and retain very little intelligibility with modern Hochdeutsch.
For comparison, here's the Lord's prayer in modern Standard German (which itself seems a little archaic to modern German ears):
"Vater unser im Himmel, geheiligt werde dein Name; dein Reich komme; dein Wille geschehe, wie im Himmel so auf Erden. Unser tägliches Brot gib uns heute. Und vergib uns unsere Schuld, wie auch wir vergeben unsern Schuldigern; und führe uns nicht in Versuchung, sondern erlöse uns von dem Bösen."
and in the East Franconian dialect of Old High German:
"Fater unser, thū thār bist in himile, sī geheilagōt thīn namo, queme thīn rīhhi, sī thīn uuillo, sō her in himile ist, sō sī her in erdu, unsar brōt tagalīhhaz gib uns hiutu, inti furlāz uns unsara sculdi sō uuir furlāzemēs unsarēn sculdīgōn, inti ni gileitēst unsih in costunga, ūzouh arlōsi unsih fon ubile."
(if you Babel-fished that literally word-for-word back into modern but very broken German you would get something like "Vater unser, du der bist im Himmel, sei geheiligt dein Name, komme dein Reich, sei deine Wille, so er im Himmel ist, so sei er in Erde, unser Brot tägliches gib uns heute, und verlass uns unsere Schuld, so wir verlassen unsern Schuldigern, und nicht gleitest uns in (Versuchung?), (sondern?) erlöse uns vom Übel.")
However, you also have Old Frankish which was ancestral to modern-day standard Dutch as well as many of the German dialects spoken in the Rhineland, and Old Low German which was ancestral to most of the dialects spoken in the North of Germany.
I can't speak for the other languages in much detail, but yes, all languages change over time and there will be older versions of all the languages you mentioned.
/u/keyilan knows a lot about linguistics in East Asia! Hopefully (s)he will have time for your question soon :D