> Tekst nú orusta með þeim
> A battle got under way with them

That's right, 'takask' "to begin; to take place" the middle voice form
of 'taka'.

> Þeir börðust nafnar
> They were fighting (this must mean those two Kolskeggs)

Yes, 'þeir nafnar' "the namesakes" (the two people with the same
name). sg. 'nafni' "namesake".

> og vildi eg að þú flyttir mig til frænda minna

> að þu flyttir - I thought this was use of the subjunctive and the að

served the same service as the ut in Latin where they do not use the
infinitive as we say - I ask - I strive to - in Latin they ask - or
even command THAT - something should happen
Does this work the same with Old Norse
and would the mnemonic still work with O/N
with ask command advise and strive by ut (or in O/N að) translate that
infinitive

It works with lots of verbs like this 'vilja', 'biðja' ...

Ek bið, at þú gerir mér kunnigt, hversu fólk yðvart heitir.
Please tell me what your people are called.

But the infinitive can be used in such expressions too:

ok bað hann blása
and bade him blow

En ek vil biðja þik, at þú hafir þrótt við.
I want to urge you to bear up under this.

I don't know enough to say if there are rules or preferrences
governing which type of expression is used with which verbs or types
of verbs. A couple of examples with verbs of striving/attempting.

... + inf.

þá laitaði Kálfr at flytja broeðr sína ofan til skips
then K. tried to move his brother down to the ship

ok vill nú reyna at brjóta öll heilræði föður síns
and wants now to try to break all the good advice of his father

Is it a rule that they would take 'ef' (rather than 'at') +
subjunctive, when they don't take the infinitive?

ef + subj.

ok leitar ef hann yrði varr við hvar þeir Kolbjörn lægi.
and tries to see if he could discover where Kolbjörn and the others
were lying