Most of the cards in the Core set Stark deck, which have reference to "stalwart", virtually can’t be used in that deck, because only one card (Eddard Stark) has stalwart.
Such cards as:
- Kings in the North
-Lethal Counterattack
are completely useless,

and Summer,Nymeria,Ice attachments useless in most cases.

Besides, wordings such as on Summer : "If attached character is Bran Stark, he gains stealth." make no sense because referenced characters have no stalwart.

So what was the purpose to include such unplayable cards in the deck, or may be I misunderstand something?

For sure, you are missing something. ;-) Unfortunately i can´t exactly work out what you are missing.

So i´ll just try another stalwart explanation. Basically "Stalwart" is a keyword, other keywords are e.g. "stealth" (see Tyrion Lannister, L 38), "deadly" (Ser Jaime L 37) .... Stalwart just means, if a card with the stalwart keyword is discarded or killed, put it on top of your draw deck instead of placing it in the dead pile or discard pile. The Core Set doesn´t include cards that refer specifically to "Stalwart" cards, an example for such a card is Midnight sentry, which can be found in "the battle of ruby ford" chapter pack, see: www.tzumainn.com/agot/cards/card.php .

Is it possible that you are misinterpreting the house Stark symbol in the gametext of the cards you mentioned as a abbreviation for "stalwart" ? The "direwolf head" symbol in the game text is an abbreviation for "House Stark". So if a charachter has the direwolf symbol in the gametext it means that the card mentioned in the gametext must be a Stark charachter. You get to know the house affiliation of a charachter from the symbol in the upper right corner of the charachter card.

Old Ben said:

This is almost certainly the issue. It has come up repeatedly since the release of the Core Set because in the Core Set rules, the first time the "wolf head" symbol appears is by Stalwart in the definition of the House-specific keywords. It should show up on pg. 3 of the rules where the House shields are first shown. It did for years of CCG rules documents.

So the "wolf head" means "Stark," not "Stalwart." The same is true for the lion head, stag head, squid, dragon head and sun symbols. They stand for the respective House, not the respective House-specific keyword.

Suddenly those cards don't seem so useless, huh?

Indeed/Thanks.

I must add, that in the Core set rules, there is no definition of "wolf head" as House Stark symbol at all, which is the reason of misinterpretings.

At first I thought that it is an issue of my Russian version of rules, but the same problem exists in English variant too. Perhaps, FFG need include definitions of House symbols to the official LCG FAQ.

Again, thanks for clarification Old Ben and ktom.