We have $n+n^2+\ldots+n^7=254.$ it is obvious that $n\neq1,$ so this is equivalent to saying $\dfrac{n^8-1}{n-1}-1=254\Rightarrow n^8=255n-254.$ By inspection, we see that $n=2$ is a solution, and the Rational Root Theorem implies that there are no other positive integer solutions.

Teddy Day was the fourth day, so Trevor gave his Valentine $2^4=\boxed{16}$ teddies.

On 13th Feb, the (interactions) will be n^7, clearly, n has to be <3 , because 3^7 is way too large. And it has to be an integer, and it cannot be 1 too, so only option left is 2. Which on verification yields the result 254.
Now required thing is teddies, which will be 2^4 = 16.

P.S. That lucky guy and the lucky day (Feb 13) :p

$\text{Roses} + \text{Proposals} + \text{Chocolates} + \text{Teddies} + \text{Promises} + \text{Hugs} + \text{Kisses} = 254$

$n +n^2 + n^3 + n^4 + n^5 + n^6 + n^7 = 254$

$n > 0 \therefore n^7 < 254$

$n < \sqrt[7]{254}$

$n < 2.2057\cdots$

$n = 1 \text{ or } 2$

$\text{If } n = 1,~ n + n^2 + n^3 + n^4 + n^5 + n^6 + n^7 = 7 \neq 254$

$n \neq 1 \therefore n = 2$

$\text{Presents given on day } x = n^x$

$\text{Teddies are given on day } 4 \therefore x = 4$

$\text{Presents given on day } 4 = 2^4 = 16$

$\large \boxed{\text{Teddies given} = 16}$

On factorizing 254 we get 2x127. And we know that total no of interactions is n^1+n^2...n^7. Which can be written as n(1+n+n²+...n^6). As we know 2 multiples are n and 1+n+n²... either one is two. The latter one cannot be 2 as the value of n will not be a positive integer then thus we can conclude n is 2 And the number of teddies is 2⁴=16