Tension in Ideal Strings I know that elastic strings have zero tension $T$, when it has it's natural length $L$, and that they experience tension when there is some elongation.
In the case of ideal strings, which are inextensible, is the value of tension always fixed for it's length never changes? 
Also can ideal strings be bent since they will still have the same length which complies with the assumption of constant length with time?
I would greatly appreciate if anyone of you could answer my querry since I am currently preparing for a competitive examination and I require some conceptual clarity beyond the textbook.

 A: 
...is the value of tension always fixed for it's length never changes?

No. The tension in ideal strings is not at all related to the extension, because their length is fixed. To get a feel for this, imagine a normal elastic string which is quite stretchable. Now slowly increase the "stiffness" i.e. rigidity of this string. So, initially if a force of $5N$ would stretch the string by $2cm$, then now the string will get stretched by an amount less than $2cm$ for the same $5N$ force. Thus when you keep on increasing the "stiffness" forever, then you will approach an ideal string. So an ideal string doesn't stretch at all for any amount of force(assuming that it is unbreakable too).

Also can ideal strings be bent since they will still have the same length which complies with the assumption of constant length with time?

Yes. There is no problem in changing the shape of an ideal string in any way as long as you don't stretch(or shrink) it. If you weren't able to bend an ideal string, then how would you get it around a pulley?!(you will encounter many questions about the Atwood machine if you haven't yet, and these all won't have existed if an ideal string could not be bent).

Will $T'=T$?

A big NO. Since ideal strings are massless, so any massless element of the string cannot have any net forces acting on it, or else it will have infinite acceleration which is physically not possible($F=ma \Rightarrow a=\frac{F}{m} $, so $m=0$ & $F\neq 0 \Rightarrow a=\infty$). So if you try to balance forces at the end where the force is being applied, then you will see for yourself that the tension has to be equal to the applied force. Even if the wall starts acclerating due to the force/tension applied by the string, still the tension in the string will be equal to the applied force. You can again verify this by drawing the free body diagram of the point where the external force is being applied.
Food for thought :- Try to prove that the tension along an ideal string is contant or same everywhere on the string. Make use of the fact that an ideal string is massless.
