The commentators all note the significant place that the root š.ʾ.l has in the opening chapter of Samuel. Because of the punch-line ("he is lent (šāʾûl) to the LORD") in v. 28, many also are minded to think that it belongs, rather, to a now missing birth narrative of Saul. This remains contentious.
It is also the case that the forms that bother OP have bothered commentators for a long, long time. In the 11th C, Rashi saw the need of providing a clear gloss for v. 28 in particular.
(1) How do we arrive at "I have lent" and "he is lent"? Is there support elsewhere for these conjugations with this meaning?
A fine question. The fact that dictionaries like Brown-Driver-Briggs give "lend" as one of the definitions, doesn't explain the oddity of getting from a Qal meaning of "ask" to a different concept altogether.
The "normal" explanation goes something like this. We have, in v. 28, the Hiphil form of the verb (hišʾiltîhû), as OP rightly notes. There are only two occurrences of this otherwise common verb (>160x in the Qal) in the Hebrew Bible: this one, and Exodus 12:36. Of those two, only this example in 1 Sam 1:28 must be Hiphil according to the consonantal text; some re-vocalize the Exodus text to get a Qal.
Given those two texts, though, the suggestion goes that the Hiphil means to "let a person ask", so idiomatically "grant a request", and so here, literally, as S.R. Driver has it, "let (one) ask him for YHWH" = "let him be asked for (lent him to) YHWH".
This doesn't seem very satisfactory. Others, then, suggest that the Hiphil carries the connotation of "dedicate". Noting the combination with the preposition le-, David Tsumura explains the sense of the Hiphil this way:
A causes X to be requested to B = A entrusted X to B
The level of uncertainty about either of these explanations (both of which have their followers and detractors in the wider literature) can be seen in the entry for HALOT which has difficulty seeing that either of them have much to commend them (see the conclusion in
(c) in the entry):
1S 128: there are two possible interpretations: (a) traditionally to lend, loan, so e.g. Gesenius-Buhl Handw.; Zorell Lexicon; Stoebe 1. Sam. 99; ZüBi; NRSV: I have lent him to the Lord; cf. MHeb. שָׁאַל hif., JArm. שְׁאַל af. to loan out; so also Nab. (Lidzbarsky Handbuch 371; Cantineau Nabatéen 2: 148a); (b) “I treat him as one who has been requested ליהוה from Yahweh”, i.e. I entrust him (or consecrate him) to Yahweh, so Westermann Ges. St. 2: 17413; cf. TOB: I hand him over to Yahweh, assign him to Yahweh; REB: I make him over to the Lord; (c) deciding between these two possible suggestions is not easy, for in the end neither interpretation is very far removed from the other; the evidence from the cognate languages perhaps gives precedence to the first of them (a).
Perhaps, as Graeme Auld points out, the language seemed as tortured to the first readers as it does to us today, and is simply the author making a bad pun, or using forced language in order to keep up the use of the key root, š.ʾ.l.
(2) Is "lent to the LORD" a known way to describe dedication to temple service like this?
Not to me. This language of "lending" is so odd in any case, that I'm not sure it's worth searching. I would be interested to know if others might be aware of something here, though.