出ていってのいって

[MeitanteiJesus]

Is the いって in 出ていって, 行って？

If so, for something like: ゆっくりしていって

What function does the して serve?

I assume it translates as lets take it easy / go slowly, so wouldn't the して make it slowly do/go? Or is it more like Lets go and do it slowly?

Also, is it frequent that kanji like 行 and 来 is written in Hiragana when attached to another kanji?

[Ongakuka]

Is the いって in 出ていって, 行って？

Yes

I assume it translates as lets take it easy

That's right!

What function does the して serve?

Remember, there's no object in this phrase. It's not ゆっくり（何かを）していって so the even if we translate して as do, there is no specific action that comes to mind.

If you say 'Take it easy' what exactly are you taking?

Also, is it frequent that kanji like 行 and 来 is written in Hiragana when attached to another kanji?

Yes that's right.

Apologies if I have said anything inaccurate here

[richvh]

Note that the いく in 出ていく and in していく serve different purposes. In 出ていく it denotes physical motion (leave and go = depart), whereas in していく it's motion in time: do (something) proceeding forward in time. Reference: the last part of Other uses of the te-form.