To Q1.

Given a effective Hodge structure $V = \oplus_{p+q=k} V^{p,q} $ of weight $k$ one can define the $m$-th Tate-Twist $V(m)^{p,q} = V^{p+m,q+m}$. This a Hodge structure of weight $k-2m$. Note that in general $m$ can be any integer. But restricting to natural numbers, there will be a $m$ which produces a twist that makes $V(m)$ vanish , since effective means that for $p,q$ $< 0$ every $V^{p,q}$ vanishes and we assumed $V$ to be effective.

That means that Tate twisting by natural numbers induces some kind of torsion the "on" Hodge structures. At least this is how i understand it.

In the category of effective Chow-Motive there is also Tate-Twisting by the Tate-Motive.
Some authors have the Lefschetz-Motive as one contained in the decomposition of $\mathbb{P}^1$ and define the Tate-Motive as its formal inverse to generate the (general) Chow-Motives,while others do it vice versa like Manin in his 1968 paper "CORRESPONDENCES, MOTIFS AND MONOIDAL TRANSFORMATIONS".

In the Chow-Motives,which are triples $(X,\rho,n)$,while $\rho$ is a projector on $X$ and $n$ its degree, there is the following rule:

$(X,\rho,n) \otimes (y,\pi,m) = (X\bigsqcup Y,\rho \times \pi,n+m)$.

The Chow-Motives contain the effective Chow-Motives as subcategory. Lets agree that the Tate-Motive is the one having negative degree. Then Twisting by the Tate-Motive is lowering the degree of every Motive in the Chow-Motives. A motive that is embedded into this category by being also a effective Motive, would in this case become none effective.

In both cases the twisting somehow extends the set of objects that are looked on to a less special case,that might enable new results.