Your basic assumptions are correct.
It can be observed as such:
- In Q the carbocation is stablised by resonance, inductive and hyperconjugative effects so it is quite stable and will be formed fastest.
- Comparing P and R: In P, a secondary carbocation is formed and there are 4+1 or 5 hyperconjugative structures possible. Moreover, the inductive effect is also greater in P than in R. Hence, it forms carbocation faster.
- Both R and S form primary carbocations. Inductive effect is more prevelent in R than in S, while S has 3 hyperconjugating structures.
But R offers capacity to rearrange and stablise the carbocation, by methyl migration to produce highly stable tertiary carbocation, also stablised by resonance over the adjoining phenyl ring.
But, S also forms a carbocation which is primary, but offers no stablising effect other than 2 +1 or 3 hyperconjugating structures.
Similarly, this rearrangement may also be attributed to the comparison of P and R, where hydride shift and methyl shift occur. The migration amplitude of hydride, being lower, leads to rearrangement of positive charge quickly and hence is kinetically also favoured.
Thus finally the order may be: