Q: Is Basque truly unrelated to any other Terran language?
A: (note: usually, linguists use the term "human language", unless they also mean unrelated to zoolinguistic communication forms as well, which generally isn’t talked about) Nobody knows for sure. Chances are, it is related to some, or all human languages, but the time of divergence is so far back that we have no way of knowing. A good hint would be genetics, but then human migration patterns don’t always fit linguistic events.
Q: Is Welsh also unrelated to any other Terran Language?
A: No, it actually is related to other human languages (confirmed), specifically the Celtic branch of the Indo-European languages:
/—-Welsh
/—————<
/————Brythonic—| \—-Breton
| \—Cornish
Celtic-|—Insular(extinct)
| /————-Manx
\————Goidelic——| /—-Irish
\——-Gaelic—<
\—-Scottish
Q: Is Hungarian, (which is not related to any language in the same geographical location), actually related to the Semetic-Asian language group?
A: Perhaps you mean the Afro-Asiatic language family (or the "Semito-Hamitic" language family)? Well, it might be, but as far back as we can trace, nobody has found any concrete relationship. In other words, no. Hungarian is from the Urgic sub-branch of the Finno-Urgic language family, sometimes grouped together with the Samoyedic language family (making the Uralic language family, sometimes grouped with the Altaic language family to make the Ural-Altaic language family. but we have no concrete linking of Uralic and Altaic, just some grammatical and phonetic indications)
Q: The modern Russian alphabet seems derived from the Greek; are the Semetic letters (Ahmaric, Hebrew, Persian, Arabic) all also derived from a common source?
Generally it is called the Cyrillic alphabet, as it’s not just Russian, but a whole slew of other languages both related and unrelated to Russian use it. The alphabet was invented by two missionaries, and your judgement is correct, most of the letters were taken from Greek, and a couple from Hebrew.
But about "Semitic" letters: the Amharic alphabet was invented fairly recently, perhaps you mean Aramaic? The Aramaic alphabet and the Hebrew alphabet are very closely related. The Persian and Arabic alphabets are one and the same (but one should note that Persian is not Semitic, even though Arabic is. Persian is Indo-European), and yes, the Perso-Arabic script (as many like to call it) is related to the Hebrew-Aramaic script. If you look at the letters, you will often find similarities if you look hard enough. For example, the the Hebrew letter "Siin" looks exactly the same as the Arabic letter of the same name. Of course, Arabic letters have 4 forms each, depending on whether they are at the beginning, middle, or end of a word, or if they are alone, so sometimes they will not look exactly the same, but if the Arabic "Siin" is at the beginning or middle of a word, it will look just like Hebrew, and if it is at the end or alone, it will look just like Hebrew, except with a tail at the end. Then there is "Shin", in Hebrew, you simply add a dot to "Siin", and in Arabic, you add a caret (or three dots) above Siin to get "Shin".
Agoraphile is incorrect, not all of the worlds writing systems are related, although most of them fall into one megafamily, starting (I think?) with the first writing system ever, Sumerian cuneiform. This developed into the modern mega-subfamilies of Semitic and Brahmic, Brahmic includes all the scripts of India, as well as the scripts of Cambodia, Laos, and the indigenous scripts of the Philippines (the subfamily is named this way because all these scripts descend from a single script, the Brahmi script). The Semitic includes Hebrew, Arabic, Greek, Latin, Cyrillic, and perhaps a couple others. Arabic itself has one daughter script, called Divehi, used in the Maldives.
Q: If Semetic-Asian languages can be grouped together, how are Mandarin and Hebrew related?
A: Earlier I said that perhaps you mean Afro-Asiatic languages. To the best of our knowledge, Afro-Asiatic languages are only spoken in North Africa and the Middle East, so they definitely do not include Mandarin.
I apologise if I sound condescending, I don’t mean it.