Hm, so many questions...
0.0 - What is the purpose of Ti role?
"The individual is able to talk about things from a dispassionate academic or theoretical point of view for brief periods of time, but seems overly bookish when doing so and tends to grows tense."
In other words, they suddenly fulfill their role as a teacher archetype in the society. And as a purpose, this means they tends to be theoretical and oriented to academic at one point, but when they get criticized, they either dismiss it or take it to heart. However, this role is necessarily needed to cover their inconsistency in judgment - not to mention they have almost a non-existent Te - which also is alluded into their ethics. Furthermore, role function, as the name suggests, is a roleplaying function.
1.1 - what the point of Ti role in EII is?
It's not easy for EII to apply LSI's cold logic fully, but they need this element to function for adding more consistency to their judgment.
1.2 - How can it be / be made to be a useful thing?
Ti role's usage is mostly related to their Fi, which means they apply logic to manipulate facts and preserve the current relationship.
2.1 - could Ti tie in with Fi somehow without having to end up needing SE and *without* violating Fi?
I don't really understand the question but if I were to assume, each element works to each other (this is why it's called IME) since it's more about metabolism process than being an independent element to function. So, for example, Ti in EII could tie with Fi, but also need Se in the process without violating or, in other words, contradicting their Fi's ethical principle as well.
2.2 - Does Ti role in ESI work any differently than in EII?
Yes. and if EII uses LSI's Ti, ESI uses LII's Ti but still, it's weaker by usage akin to their Super-Ego relation.