Valuable thought. .Great update. I'm looking forward to more.
One thing that might strain future Swedo-Russian relations are the serfs. OTL, some Russian serfs were espaping to the Polish-Lithuanian Commonwealth int he 17th century due to the slightly better conditions there.
Karl XI tried to get serfdom abolished in Estonia and Livonia,but the motion was defeated in the parliament of both Duchies - he abolished it by decree in Ingria and Kexholm (not that there were many sefs in either province, but still). However, the reduction of noble held land in Estonia and Livonia meant a de facto if not de jure abolishment of serfdom - the Swedish crown freed the serfs it got from seized land, reintroduced the moving week (giving the serfs the right to choose another landlord and move from their current to their new one during one week of the year) and abolished the Estonian and Livonian nobility's right to "hand and neck", ie being police, prosecutor, judge and executioner all in one on their own estate, putting the legal rights in the hands of the Swedish governors and in theory (practise was something else entirely, as we all know) gave the Estonian and Livonian serfs equal rights before the law.
Estonian and Livonian nobility quickly had to improve their treatment of their serfs, in essence making them tenants, since otherwise they would all move to crown estates where they were treated better.
Some of the serfs and working force Peter moves to Ingria will pick up on the difference in treatment of their social class across the border and might escape, and Peter and/or the Russian nobility that "owns" these serfs will be wanting them back. I'm sure Karl XII will be happy to return most of them in the interest of good relations, but sooner or later there will be some case where Sweden refuses - perhaps because the serf has joined a Swedish garrison regiment and sworn an oath to the Swedish king and is in theory a Swedish subject now.
Karl might also do a shake-up of the government system of Estonia and Livonia - the nobility that would side with an outsider probably tried during the siege of Riga and can have their estates and title of nobility revoked, which might persuade the others in the parliaments to support a change where Karl XII can rule by decree as he can in Sweden, or at least a formal abolishment of serfdom.
Regardless, with the increased trade, there is bound to be new wealth created in Russia and with it some social change. Many of the goods Russia sell (tar, hemp, lumber) are "cash crops" made by the serfs, and there were historically small trade empires created by serfs who either bought their freedom or just hired others to deliver their due (usually day labour) to their landlords. With the earlier increased trade, there are bound to be more and earlier serf trade companies, and through them information will spread.
It will be interesting to see if this becomes a factor ITTL.
AFAIK, in OTL Russia the serfs had been trying to escape in all directions: the PLC, Don, Sich, Ottoman Empire. The problem (for them) was that for most of them it was a long way to go with a very good chance to be intercepted, especially if they were trying to flee with the families and at least some movable property. With the Don government, IIRC, eventually came to an arrangement regarding their return (breaking the old tradition “there is no return from the Don”): as soon as status of the Cossacks of Don had been changed from the marginally tolerated bandits to the privileged servants of the crown (and the birder moved further to the South) their attitudes changed leaving the openings in the hosts of the border areas, etc.
Escaping into the PLC was easier: peasants in the Belorussian and Ukrainian areas used a similar language and were mostly Orthodox. Not the case with the Finns and Estonians. Of course, escapes would not be fully prevented but they would be difficult, especially if there is a lot of the low level trade is happening across the border and these peasants are not interested in strained relations with the Russian authorities.
Now, regarding the potential Russian-Swedish official issue on that account, well, most probably there would be an agreement attending to most of the cases so this should not be a major issue and in the situation you described an escaping serf simply would not be permitted to join a garrison: such a person would be easily identified by inability to communicate in German, Swedish, Finnish or Estonian. 😜 But surely handling such cases would be considered and properly documented: people of that period had been very good when it was coming to the bureaucracy.
As far as the rich serfs are involved, there were, indeed the very rich serfs engaged in a trade but you are slightly off regarding their relations with the owners. They were not doing “the daily work” (barschina) but going on “obrok”: agreed upon cash payments and perhaps some additional services (if the owner lived in his estate and a serf was operating in a city then perhaps some purchases or even contacts with the bureaucracy). They could not become members of the merchant guilds unless liberated by their owner (for payment or some valuable service) but other than that there was no restrictions to their activities. In the late XVIII - early XIX Count Sheremetev, one of the richest people in Russia (who scandalized CII by asking permission to marry his former serf, a star of his domestic theater, it took a while but eventually they did get married), considered it a some kind of amusement to have serfs-millionaries (probably simply very rich): he was not restricting their commercial activities but refused to accept pay off money. One of them got his freedom for being in a right time in a right place with a right stuff. He brough a barrel of oysters as a present to his master when the count, who was giving a banquet to his friends, found that he run out of oysters. Upon seeing the serf in a parlor he made him an offer: oysters ASAP for freedom, the serf immediately returned with a barrel on which his emancipation document had been signed.
But the less affluent owners would accept cash more eagerly because the Russian nobility was “systematically” short of it. So these emancipated serfs will join the merchant guilds and you know what would be their first desire? To get permission to own the serfs. CII bumped into that problem during the work of her Commission: the merchants asked for this right because they considered serfs more reliable than the salaried employees. This was XVIII Russia with its attitudes.