1. Why is the verb 狭{せば}む so rare/weird?
As user naruto said in the comments, the reason you don't see 狭{せば}む much in modern Japanese (and that your input method can't handle it) is that this is a Classical Japanese (bungo 文語) verb, of the shimo-nidan conjugation; it inflects as 狭め(ぬ), 狭め(て), 狭む(ぞ), 狭むる(人), 狭むれ(ど), 狭めよ(!). (If you want to learn more, you can try a Classical Japanese grammar). In Modern Japanese, shimo-nidan verbs like, say, 求{もと}む got a shimo-ichidan form like 求{もと}める, so in the case of Classical 狭{せば}む we can predict modern 狭{せば}める, which is indeed a fairly common verb (108k hits in Google Books for 狭め). And there's also the intransitive variant, 狭まる, which works as a modern verb too.
As for the rarity of the kanji, both ‹挟› and ‹狭› are part of the Jōyō Kanji table (they're shown right next to each other, in fact), meaning we are supposed to learn them; the Jōyō readings for them (i.e. the readings that have to be learned) are:
- 挟：キョウ、はさむ、はさまる
- 狭：キョウ、せまい、せばまる、せばめる
狭{せま}い is undoubtedly the most common reading of ‹狭›, and you should definitly learn it and take care to distinguish from ‹挟›.
Now for your main question, which can be subdivided into two distinct questions:
2. Is there a relationship between the similar-looking kanji 挟 and 狭?
Questions about kanji are properly questions about Chinese.
In Mandarin these are read as xié and xiá, which sounds very similar (and in Japanese on'yomi, which comes from old Chinese, they're identical: kyō). They also share a component, which in the old form is written as 夾, and in even older forms is clearly a pictograph of a person between two other things, possibly two people, perhaps holding them with their arms:
Used by itself, 夾 also sounds similar: jiā. What's more, the meanings of the three words all seem related. This is usually a hint of an etymological word family. Let's consult Schuessler's etymological dictionary:
- 夾 jiā < *krêp: press between, be on both sides, support.
- 狹 xiá < *ɦkrêp: narrow.
- 挾 xié < *gêp?: grasp under the arm, hold onto, conform.
There are a bunch of complications, but for our purposes what I wanted to show you is that there were a bunch of similar Chinese words, to which they ascribed the same basic character (夾), and then distinguished the various nuances by adding determiners (stuff like 扌 or ⺨). Sometimes they added the determiners quite willy-nilly, and in the earlier stages of Chinese writing they were kinda optional and in flux. This process was very productive in general—in the case of 夾 we also have things like 峡 xiá gorge, ravine (also used in Japanese); 鋏 jiá tongs (sometimes seen in Japanese for 'scissors', hasami); 梜 jiá chopsticks (now archaic); 浹 jiā soak; 陜 xiá variant of 狹…
3. Is there a relationship between the Japanese words sebamu, sebameru, semai, hasamu, hasamaru?
To answer this we first have to set aside kanji and think of Japanese itself (it's always good to keep in mind that Japanese has an existence independent of Chinese writing).
We've seen how sebameru is a regular derivation of old sebamu, 'to narrow; to cause grief'. And sebamaru is the intransitive form, 'to become narrow' (like tomeru/tomaru).
It's very easy to connect the adjective semai to these verbs, via a root seb-/sem-. The consonants /b/ and /m/ are quite similar; both are bilabials (made by closing both lips), distinguished only by a nasal sound in /m/. What's more, /b/ is thought to have sounded like 'mb' in Old Japanese (as it still does in e.g. Tōhoku); 'mb' and 'm' are even closer, and there's a number of other b/m alternations in the Japanese lexicon. And of course the meaning is basically the same, changing only from adjectival to verbal forms.
What about hasamu, hasamaru? Well, they're not nearly as similar, and the meaning, 'to interpose, to insert between; to be on either side; to get between', while related, isn't quite the same. seb-/sem- is about narrowness, while hasam- is about in-between-ness.
Now think from the point of view of the Japanese scribes and monks who chose which kanji to use for which Japanese word—which is to say, who translated between Chinese and Japanese. They clearly decided to translate hasam- as krêp 挾, 'press between' (one might expect the original 夾 character for this; but in many cases the 'derived' characters like 挾 still could be used for the 'source' words like krêp/夾, and I bet this was the case here). And they decided to translate the sem-/seb- family, 'narrow', logically enough, as the Chinese adjective 'narrow', 狹 ɦkrêp. At the time both of them sounded like kep, which is how their got their on-yomi; -ep in Japanese (in this as in other words) changed into -epu > -efu > -eu > -yō, making for modern kyō.
At some point some genius also decided to write hasami, "scissors", which is just 挟み, with the 鋏 character, on the grounds of it being the same basic 夾 but having the 'metal' 金 determiner—despite the fact that in Chinese the character actually meant 'metal tongs'.
