If I cast Redirect on a some kind of counter spell can I choose the Redirect as new target for the counterspell?
Yes, you can select Redirect as the target of the counterspell.
The Gatherer ruling that BJ Myers quoted sums up the matter nicely, but if you want to see the Comp Rules that the ruling is based on, here they are.
A counterspell cannot be redirected to target itself based on rule 114.5:
114.5. A spell or ability on the stack is an illegal target for itself.
The rules that allow Redirect to Redirect a spell to target Redirect are in 608 (Resolving Spells and Abilities):
608.2c The controller of the spell or ability follows its instructions in the order written.
608.2k As the final part of an instant or sorcery spell's resolution, the spell is put into its owner's graveyard. As the final part of an ability's resolution, the ability is removed from the stack and ceases to exist.
Since the spell isn't removed from the stack until after its resolution is complete, it is still available as a target during the resolution itself.
And if you're wondering why spells aren't allowed to target themselves, look no farther than Redirect itself. If Redirect was targeting itself, what would happen? Would it change its own target to a different spell, and then change that target (effectively changing two targets)? Would it change its own target, but then not do anything to the spell it's targeting?
The implications are weird, so they solved the problem with a blanket ban.
Yes. From the Gatherer rulings:
If you cast Redirect targeting a spell that targets a spell on the stack (like Cancel does, for example), you can’t change that spell’s target to itself. You can, however, change that spell’s target to Redirect. If you do, that spell won’t resolve when it tries to resolve because Redirect will have left the stack by then.