Does padding with dummy bits allow an NP-problem to be solved in fast exponential time?

Take this example mentioned here: NP-hard problems with very fast exponential-time algorithms

We can create such problem by padding assuming ETH‌. Take an NP-complete problem $$L$$ such that $$L$$ is decidable in time $$O(2^n)$$, by padding $$L$$ with some dummy 1's, create $$L'=\{1^{n−(\log_21.01)n}x:|x|=(\log_21.01)n∧x∈L\}$$. it is easy to prove that $$L'$$ is complete for NP and the running time of $$L'$$ is exactly $$O(1.01^n)$$.

Because I'm a newbie in mathematics and computer science, I don't quite understand this answer. Does it mean that padding an NP-problem that can be solved in $$O(2^n)$$ with a string of dummies 1, we can construct a new variant of the original problem that can be solved in fast exponential time $$O(1.01^n)$$?

As an additional question: can I apply this technique to Levin's universal search (http://www.scholarpedia.org/article/Universal_search)?

Here is a more extreme example: $$\mathrm{SAT_{PAD}} = \{1^{2^n} 0 \phi : \text{\phi is a satisfiable CNF on n variables}\}.$$ This language is decidable in polynomial time.
What padding accomplishes is it constructs a language in a lower complexity class; it doesn't actually help you solve the original problem. All it does is artificially modify the parameter $$n$$.
Running an algorithm for $$L$$ on the language $$L'$$ results in a running time which is $$O(1.01^n)$$, where $$n$$ is the size of the padded instance; the running time is still $$O(2^n)$$ in terms of the size of the original instance.