No, it would not necessarily require a trellis. I have 3 growing without a trellis and they are bigger than a load of Buicks.

I would need to see the foliage, see the exact bloom very close up and need to know how often it blooms, etc., before I could tell you what your rose is. Thousands of roses look similar and pink is the signature rose color. From a distance, it does look like a polyantha, china or noisette. It could even be an old tea rose (not to be mistaken with hybrid tea). Yes, it does appear to be an old garden rose. Although I could be and have been mistaken before.