If he went. Why wouldn't he go? It was free and he was the son of a socially ambitious man. There's no compelling reason for him NOT to have gone. Being lame and disgraced has nothing to do with age and Shakespeare could have been or he could have been speak metaphorically. He also was known to play older men and in comparison with a teenager a 30 year old can seem ancient. Again, old can also be metaphorical. In what sonnets does De Vere implore Southampton to marry his daughter? The first 17 sonnets ask the young man to procreate that's it. No specific mention of anyone by name. The only specific mention of a name is Will in sonnet 135 and 136. Nobody by the name Eddie and don't give the whole drivel about a pseudonym for which there is no evidence.
The learning in the plays primarily is derived from the source story used. Whole passages are lifted from Holished's "Chronicles" and Plutarch "Lives". So the learning or being an expert on a subject will come from whoever wrote the source not necessarily the author's experience. Can you prove that De Vere owned ANY of the sources used in writing the plays? Or do you just assume he knew because he was the all powerful De Vere?
"Would a commoner even be allowed to write to Aristocracy in those days?" Yes, if the aristocrat allowed it. Its the same as being friends with someone famous or a big boss today. If they're good with a certain level of treatment then that's the way you interact with them but not everyone can act like that around them. Its assuming a lot that Shakespeare had this level of intimacy but no more so than speculating that De Vere and Southampton had one.