Can the Dalish read the common tongue?

I don't see why they wouldn't be able to, provided they can read at all (and I don't see much reason, for that matter, to assume they can't). From what we've seen, they don't really speak Elvish, they just kind of pepper their speech with it, Super-Friends style; the language itself is still mostly dead and there's nobody out there who knows enough of it to speak it fluently anyway.

Considering there is no longer said to be uniformity among the clans, I would imagine it would entirely depend on the Keeper and other clan elders whether they thought teaching them the common script was going to be of use. An isolationist clan that wanted nothing to do with their human neighbours probably wouldn't bother. A clan which freely trades with them would likely think it something useful for every clan member to know.   Luckily Lavellan came from a clan that wasn't adverse to peaceful interaction with their neighbours, so had been taught enough common script to get by.   Since you can be a warrior or hunter as well as the First of the clan, the Keeper would hardly send you on a spying mission if you would be unable to read important information along the way.

I think every clan member would be taught rudimentary elven runes as well. Many clans seem to actively search out old ruins in the hope of finding something useful within. If Marethari was anything to judge by, she didn't go in herself but sent other members of the clan to investigate. They would need to be able to work out if something was a warning or useful information when confronted with it, so must have been given sufficient teaching to enable this.

I think the oral traditions of the Dalish probably involve a sort of patois combining common and elven but that would be rather difficult to show in game, so they always seem to be speaking common. The reality would be that they would use elven in their speech as much as possible because of the fact that they are anxious not to lose it. However, whilst the writing often does seem to be intended for an human Inquisitor, the fact that both Solas and Morrigan can hide what they are really saying in elven from Lavellan if they choose to, would suggest that the Dalish knowledge of their own language is pretty limited. What Solas claims to say to Abelas is not the exact translation of his words based on what limited access we have been given to the language.

Well, there is clan Virnehn which is absolutely hostile to humans. The one from the Masked Empire.