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I am going through the word 奇. Mandarin Temple had mentioned, learning hint as “a person riding a horse”. Is this only learning hint? Or is it also etymologically correct? What is the exact etymology of 奇?
For "strange", can I use 奇, or is it obsolete? Must I use 奇怪 instead?
「奇」 is not a person riding a horse - that is probably a learning mnemonic, meant to remind you of the word 「騎」. This character has received some attention in the question What is the glyph origin of '奇'?.
「怪」 (Zhengzhang OC: /*kruːds/, strange) is comprised of semantic 「心・忄」 (heart, mind) and phonetic 「圣」(/*kʰuːd/).
「圣」 (Mandarin Pinyin: kū, to dig) is comprised of semantic 「又」 (picture of a hand) and semantic 「土」 (picture of a lump of dirt).
「圣」 is rarely seen in the orthodox character system. However, you may see the identical shape 「圣」 in unorthodox systems representing completely different words or components, such as Simplified Chinese (where it is used as an abbreviation of 「聖」) and Japanese Shinjitai (where it is used as an abbreviation of 「巠」).
As usual, these abbreviations do not have anything to do with the language.
You shouldn't use 「奇」 by itself if you want to say strange; this is because 「奇」 forms the roots of several words which don't necessarily mean strange, but related yet different words like wonderful (奇妙), mysterious; magical (神奇), and rare (奇特).
Tl;dr: Isolated use is largely impermissible (points 3. and 4.), because they are classical Chinese. Evidence for which is outlined in point 2.
奇 and 怪, I shall adopt another approach at your question.On 奇怪 as a compound word
Compound words (複合詞) are rarer in classical than modern Chinese, but for 奇怪, plenty of examples in classical Chinese could be found (see here). The following example is from Shiji:
閎夭之徒患之，乃求有莘氏美女、驪戎之文馬，有熊九駟。他奇怪物，因殷嬖臣費仲而獻之紂，紂大說（略）《史記·周本紀》
Hongyao and others were worried (about Count of the West's imprisonment). They then procured beauties from the Youshen clan, bright horses from Lirong, and thirty-six horses from Youxiong, together with other (equally) peculiar items, and offered them to King Zhou through Fei Zhong. King Zhou was extremely pleased . . .
This tells us 奇怪 is not a recent invention. They were placed next to each other for their similarity in meaning (近義並列). Other examples traceable to classical Chinese include: 幼弱, 審察.1
On isolated use of 奇 and 怪 in modern Chinese
Because of its classical nature, isolated use of 奇 in modern Chinese is almost nonexistent, but not impossible. You may still find it in idioms, e.g. 嘖嘖稱奇.
The same goes for 怪 (consider the idiom 少見多怪). However, more often than not you will see it on its own as a verb that means 'to blame'. That is to say, there are two senses in modern Chinese: the 奇怪；怪異；怪誕 kind (meaning strange), and the 責怪；怪罪；怪+sb. kind (meaning to blame).
Reference
1 劉承慧（2003）〈古漢語實詞的複合化〉，於《古今通塞：漢語的歷史發展》，中央研究院語言學研究所籌備處，p107–139。
奇 is a controversial one. There are sites that say it is a person on a horse being original form of 骑, but there are no old glyphs with the horse character in the ancient script.
Easiest interpretation would be large 大 and gasping 可 (mouth + breath). A person is gasping in awe at a large peculiar object. 可 can also double as phonetic, although they both don't really sound alike.
怪 is combination of heart 心 and 圣, which is a hand holding a lump of clay. This is probably an ancient form of 块 (which had another old variant form 凷). Most likely just a phonetic component, but you could think of it as an unpleasant feeling like having a lump in your throat or heart.
奇 is more neutral sounding, meaning strange but not necessarily in a bad way. It can even be used positively, being something wonderful or remarkable, like 奇迹 miracle. It is almost never used alone, possibly because it has many other homophones/similar sounding characters.
怪 sounds like ghost 鬼 and is almost always used in a negative tone. For example, monster 怪物, blame 责怪 (责 is often omitted if used in this sense), and a vulgar slang dialectal term 怪𡳞 which means annoying. 怪 can be used alone for blame or strange as it doesn't have as many common homophones.
