Who's King Roberts proxy between Jon Arryn and Eddard Stark?

Question

After Lord Jon Arryn's untimely demise, King Robert of the House Baratheon, the First of His Name, King of the Andals and the First Men, Lord of the Seven Kingdoms and Protector of the Realm, decides to make Eddard Stark, Lord of Winterfell, the new Hand of the King. He furthermore chooses to not just send a raven to summon Ned to court but travel to Winterfell with half the court, including the Queen and half her family as well as their children.

We also know that the distance between King's Landing and Winterfell is quite large, albeit the numbers are all but certain (Yoren of the Night's Watch claims it's 1,000 leagues). So for all we know, they will therefore spend quite some time on the Kingsroad going there and back again.

With the King gone and no new Hand appointed, who will rule the realm and speak justice in the name of the King? Is any of the members of the small council announced to chair the small council and to serve as the King's proxy during his absence (powers that would otherwise rest with the Hand of the King)? Is there a Hand pro tempore? There is, after all, a precedence for such a thing with Tyrion Lannister serving as acting Hand in his father's stead under King Joffrey.

It is clear that King Robert was more interested in hunting and whoring than in ruling the kingdom and therefore let the small council run the realm in his stead. Still in those days there was Jon Arryn presiding the council, a man that King Robert would trust.

Answer 1

We do not know. There is no indication in the books of who was ruling the realm while King Robert was away. So this will be speculation and logical inference.

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Robert's Small council

Small Council had following members at the time:

1. Ser Barristan Selmy, Lord Commander of Kingsguard
2. Lord Petyr Baelish, Master of Coin
3. Lord Stannis Baratheon, Master of Ships
4. Lord Renly Baratheon, Master of Laws
5. Pycelle, Grand Maester of the realm
6. Lord Varys, Master of Whisperers

Any monarch would think twice before leaving his city in hands of other men but this small council was divided and thus was of no apparent threat to the King. Selmy was too noble for court intrigue. Baelish and Varys openly despised each other. Stannis disliked everyone and was disliked by everyone. Renly was charming but he did not particularly get on with Stannis and harbored his own ambitions of wedding Robert to Lord Tyrell's daughter Margaery. Pycelle was all alone.

As already suggested, it was the Small council which ruled in Robert's presence as well so it would not have been particularly difficult for the members to act without a formal hand or King to preside their proceedings.

In any case, it is not out of question that small council was in touch with Robert. Robert did stay in castles e.g. like he did in Castle Darry on his return trip. So ravens could possibly be sent to concerning castles for requesting instructions or for informing about administration issues.

Who could have possibly presided over the Small council?

So the question arises, who would have Robert appointed to head the administration while he was away? Following would be my reasoning:

1. Ser Barristan Selmy: The lord commander was a respected figure in the seven kingdoms and wasn't a man to indulge in conspiracies and court intrigue. Robert had liked the man which is evident by his appointment to Lord Commandership of Kingsguard despite fighting on the other side. But he can be dismissed instantly because he was sent by Small Council to receive King's party on their return which means he was not the one heading the administration.
2. Lord Varys: A foreigner and low-born, Varys was not viewed favorably by the people or the Lords. There were two brothers of King serving at the small council who would find it unacceptable to serve under a low born foreigner. Not to mention, Baelish would oppose Varys' command. Besides Robert had no particular liking for the man. He pardoned Varys and used him where he could but there is no indication whether he liked him enough. So on account of low birth and low public opinion, we can dismiss Varys as well.
3. Lord Baelish: Great-Grandson of a foreign mercenary, Petyr Baelish was smallest of the small lords in the realm. Because there were better born men present on the small council, Baelish can be dismissed as well. Moreover, Varys would be opposed to his appointment. Stannis would be as well since he disliked Baelish for his support of corrupt Goldcloaks.
4. Lord Stannis Baratheon: King's younger brother never shared a bond of affection with him. He was sour and prickly and was immensely disliked by Robert. Since Stannis ran from Kingslanding as soon as Robert left, we can dismiss the Idea that Stannis could have been running the realm while Robert was away. Furthermore, Renly would be opposed to his appointment.
5. Lord Renly Baratheon: Charming and on good terms with his brother The King, Renly would be perfect for heading the realm while Robert was away. But we know he was sent with Ser Barristan to receive the Royal party from Kingslanding on council's behest so he could not be the one presiding the small council. And of course Stannis would oppose his appointment.
6. Grand Maester Pycelle: Old, wise and experienced Pycelle is what we are left with then. He had been Grand Maester for forty years and thus had more experience of statescraft than any other member. He also did not have any open quarrel with any other member. And he was allied with Queen Cersei Lannister. So it is very likely that he could be the one leading the small council in this period. Robert had no particular reason to distrust Pycelle. Not to mention, Robert often gave Cersei what she wanted as long as he could get rid of her nagging. So if Pycelle was allied with Cersei, he would have her backing as well.

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As you have mentioned, it is legally possible to appoint an acting hand of King until the official/permanent one could take his place. It is also possible that Council may have been acting as a combined body acting through popular vote.