Russ is almost certainly still alive. It has been shown that a number of members of the 13th company that went with Russ into the Eye of Terror are still alive. If the wolves still live after ten thousand years of fighting in the Eye, then Russ is more than likely still alive as well. Although the members of the 13th company that have been encountered will not say that he is still alive, they also will not say that he is dead. It is meant to be a mystery for those Wolves remaining on Fenris. They are supposed to have faith in their primarch, that he will return triumphant at the end times, reclaim his spear, and fight for the glory of the Emperor. There is much more credence to this possibility than to the one that Guilliman's wound is slowly healing in stasis. Guilliman is dead, and while it seems a nice fancy to believe that he could be healing, it is rather unlikely unless the Emperor has been able to divert some of his attention from guarding the Eternity Gate to healing Guilliman. It has been shown that on certain occasions, the Emperor does shift a portion of his consciousness away from the Gate and to his loyal subjects. On one such occasion, there was a small company of Imperial guardsmen holding off a horde of Tyranids. For 16 hours, 200 guardsmen held off millions of Tyranids until the Emperor's avenging angels, the Astartes, could come quell the invasion, without losing a single man. When asked about it later, they each replied that they saw the Emperor fighting with them and speaking with them, walking amongst them and telling them to keep fighting, that relief was coming. So it has been shown that the Emperor is able to be in many places at once, and it is possible that he could be healing Guilliman. However, I digress, back to Russ. It is, in fact, highly probably that Russ is still alive. Primarchs are hard to kill, and there are few who could match Russ in power and ability. Sanguinius is likely the only Primarch who could match Russ in martial prowess. Russ is just plain hard to kill.
One particularly interesting quote is this... "Listen closely Brothers, for my life's breath is all but spent. There shall come a time far from now when our Chapter itself is dying, even as I am now dying, and our foes shall gather to destroy us. Then my children, I shall listen for your call in whatever realm of death holds me, and come I shall, no matter what the laws of life and death forbid. At the end I will be there. For the final battle. For the Wolftime."
The veracity of this quote is without question, however, it may not be that Russ truly meant he was dying. He may have meant only that he was dying in spirit or something of that nature. What he truly meant will never be known until he returns, however, even if Russ is dead, which I think unlikely, he promises that he will return, and Russ does not make promises that he cannot keep. Even if he is dead, he will find a way to return to his Chapter.
If he is still alive, he and the 13th company still travel the Eye in search of the Thousand Sons in order to destroy them and whatever other forms of Chaos hellspawn they might encounter. It is undeniable that the 13th company is still at work in the Eye and it logically follows that Russ still runs with them. Russ' quote seems contradictory in that he claimed he was dying, and yet he took a full company with him into the Eye. It seems illogical to think that he would take a company with him to die. I personally believe that he simply meant that he was dying in soul as long as Magnus was still alive because he felt that he had failed the Emperor.
When the Space Wolves chapter has hit its lowest point, Russ will return. He will resurrect the chapter amidst fire and carnage. They will rise again, in the end times, and he will once again lead them against Chaos.