@linguistgoneforeign my Japanese teacher explained that 会う (meet) can be used with both に and と (not て). With 友達 it should not make much of a difference, in both cases it translates as “to meet with”. The subtle difference is that に indicates your movement towards your friend, whereas with と both of you might be moving towards a meeting place from different directions. However, when you are going to see a doctor 医者に会う, it should always be に as you are the person going to the doctor.
The verb forms in the sentence with て (あって\会って、して) just indicate a list of activities which then ends on a normal verb form (いきました).
The や mentioned by @jasonmolenda is 屋 as in shop, so they went to a ramen place.