In canon, isn't Pesche mostly just a comic relief character?

Yes and no. To what extent Pesche's personality as seen in chapter 247-294 is his real personality is a question I can't answer with certainty: his interactions with Nel were based on the need to hide his true identity, and to some extent his interactions with Uryuu and Renji are similar. This doesn't mean that it was all an act---it is clear from the way he talks to Uryuu during the fight with Szayel-Aporro (calling Selle Schneinder a "thingy) and his and Dondochakka's elaborate reappearance after the fight that that goofball is a part of his personality. But this isn't all of it.

Consider Pesche's experiences, as far as we know them:

• Had his mask ripped off, which is apparently an appalling act even among arrancar
• Was used as bait to trap Nel, his "master"
• Saw Nel wounded
• Was literally thrown off of Las Noches into the desert
• Saw Nel reverted into a child with no memories or powers---meaning Neliel Tu Oderschwank was, in his mind, dead.
• Decided to protect her at all costs in the brutal world of Hueco Mundo (see, for example, chapter 284)

I think, then, that Pesche has it in him to be serious. Granted, he does spend most of his "screen time", if you will, as a goof---but this doesn't mean that is his essential nature. I do regret that, at one point I characterized him as being "sad"---I feel that is not part of his character as seen in ch. 295. However, talking seriously and showing willingness to face danger and possibly death to protect a comrade are things he has done, and therefore I feel it is not unreasonable for him to do so again.

Why is Pesche so willing to tell his story, which he wouldn't even tell Nel herself?

Pesche knows that there are many arrancar in this world, and is under the assumption that people are going to assume the worst about him. He assumes people are eventually going to ask, "What's that hole thing?", and he'd rather be up-front about it, lest people think he's trying to hide something. Hence his need to explain himself. His only reason for concealing his own identity in the first place was to keep Nel's memories from returning; however, as far as he knows, her memories already have come back, so even if she were here it wouldn't be an issue, and furthermore Szayel-Aporro has realized who he is beneath his mask (as he says, "I knew you were Nel's fraccions from the moment I saw you."), so he knows the Espada won't be fooled. So, if he tells about himself, it's because there's no point to hide it anymore, and if he does hide it, it's only going to make him look suspicious.

What does Pesche think about Nel?

Nel (and Neliel) are basically Pesche's whole world. As he says in ch. 295 "We have to protect Nel-sama who is weak, even if it costs us our lives". Like all fraccions with their Espada, he is deeply loyal to her, but it goes deeper than that because of who Nel was. He would worry if she even came late to lunch. For him, her demise is probably the most important event in his life---it is how he defines himself: as the squire who failed his knight by allowing himself to be used as bait, but who will nevertheless do everything in his power to rectify his mistake, even if he dies in the process. In his life after Las Noches, Pesche's only concern in for Nel's happiness and safety, and his life mostly involves playing with her to keep her happy, and training himself to protect her better. And in the City, he often feels the need to find someone, anyone, to protect---it is the only mode of life he has known for a very long time, and it is hard to approach life any differently.

In canon, isn't Pesche stupid?

This is a real sticking point for me---Pesche is certainly not stupid. If anything, he is childish (which is not the same thing): this childishness might have always been a part of his character, but as Nel's "brother" it became his only one. To do otherwise would risk reawakening Nel's memories. In Las Noches, he keeps this up because he has to keep Uryuu and Renji from suspecting that he and Dondochakka are arrancar who used to be in Aizen's employ, as he simply wants to get Nel and get out of Las Noches as soon as possible. Also, in his interactions with Uryuu, he sees himself as being the "funny man" to Uryuu's "straight man"---he acts exaggeratedly silly and stupid to build up a funny dialog. This article on Manzai may actually make these interactions a bit clearer.

As for how Pesche speaks, in chapter 295-296, he shows that he is more than capable of speaking like an adult, and his tone is certain not of someone who is stupid. When I am having Pesche speak on things that are important to him, I try to reflect the tone in these chapters, while mixing in certain elements of his normal tone, which even in these chapters sneaks into his dialog at times.