auxilio is in the dative case. It is what is commonly called a predicative dative. This is a dative that is commonly used with the verb to be or some type of verb of sending, choosing etc. If combined with a verb of sending, choosing etc. it is most often translated as an infinitive:
nostros auxilio misit = He sent our men to help.
If it is used with the verb to be it can often still be translated with an infinitive, or simply as a noun, verb or adjective, depending on the context.
magno periculo proelium nostris erat = The battle was very dangerous for our men.
haec pecunia mihi usui est. = This money is useful to me.
Lucius erat ludibrio magno amicis. = Lucius' friends made fun of him a great deal.
hoc consilium nostrae civitati exitio erat. = This plan was the ruin of our state.
adulescenti is also in the dative, but this is a more standard dative use.
I suspect that what you think is an ut clause is not... Remember that Latin will usually begin a clause then immediately interrupt it with subclauses that provide detail/circumstance about the main clause. English will more normally express this with seperate complete sentences.
The actual ut clause is 'ut in servitutem venderetur.' = that he was sold into slavery.
This is a type of result clause, dependent on the posteaque... abiit clause.
These are commonly called a substantive clause of result 'it came about that...'
This clause is interrupted by an ablative absolute, 'delectu rite acto', then by a clause of characteristic, in this case a causal clause, 'qui militiam detractaret.'
Nor were the Tribunes of any help to the young man; and afterwards the matter played out according to custom, that a draft was held properly, and since the young man refused miilitary service he was sold into slavery.
More literally: and afterwards the matter went according to custom, that, with a draft having been done rightly, since he refused military service, he was sold into slavery.
Hope this helps...
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