The Alumni Dublinenses, register of provosts, professors, graduates and students for Trinity College Dublin, shows several separate George Alley entries as follows :
Alley, George, Pen. (Mr. McKenna), 12th August 1785 aged 17; s, (son) of Peter, Generousus; born Wicklow, B.A. Vern. 1790
Alley, George, Pen. (Mr. Staunton) 11th November 1795, aged 15, son of William, Mercator; born Dublin
Alley, George, Pen, (Mr. Newland) 4th June 1821, aged 16, son of John, Mercator, b. Dublin
Alley, George, Pen. (P.T.), 20th October 1828, aged 17, s. of George, defunctus, b. Queen's county.
The name in brackets is the tutor, the date is when the person was admitted to the college. The entries dont differentiate between city/town and County, i.e. Dublin City vs County Dublin..
Various documents such as editions of the Alumni Dublineses are usually available on the Trinity College website, but most seem to be unavailable at present as there's an update in progress. FindMyPast have an indexed version of the 1924 edition of the document.
I'm not certain what if any family details of the various Mayors might have been recorded or what may be available, I suspect most of the details relate to city council business.
To search for a baptism for your George jnr. would require Church records, so depends on which parish he was born in - I'm assuming your Alley family were Church of Ireland ?
Most of the surviving Church of Ireland records for Dublin city are available on the free IrishGenealogy website, which also has some parishes in County Dublin. Church of Ireland records for other parishes in County Dublin and other counties are not very well covered online at the moment - RootsIreland (pay-website) is often the best option, but check they have the parish(es) you require. Unfortunately early entries in the registers are not very detailed, so it's unlikely that an occupation would be included in church of Ireland records before the 1830s, address/residence would would not always be included in early registers..
A search of the Dublin city Church records on IrishGenealogy shows a number of Alley entries going back to the 1700s, and one from the 1670s, all Church of Ireland.
Townsend street would be St. Mark's parish, Fownes street St. Andrew's, and Dawson Street (location of Mansion House) St. Anne's parish.
The Representative Church Body Library is working on a project to transcribe Church of Ireland registers as part of the Anglican Records Project - they already cover some parts of County Dublin, worth keeping an eye out for further additions..
Added : I located a directory entry in Pettigrew & Oulton 1834 that might be of interest - Under the listing for Philipsburgh Avenue in the section for Clontarf there's an entry for a George Alley, esq., surgeon, at Eden Park , I dont see an entry for him in the 1840 edition.
Found a few more details that might help ... there's an entry on the Index to Dublin & Glendalough Diocese Marriage Licence Bonds for a Jane Ann Underwood and John Alley with the year 1804. The original records for these bonds do not survive, and unfortunately there's no mention of the parish - so a large area to check - Dublin city, County Dublin, Co. Wicklow etc. Luckily a summary of the marriage is included in Sir William Betham's Genealogical Abstracts, which he created before these original records were lost - the details he noted for the Marriage are as follows :
John Alley of Bishop Street Dublin,
gunmaker & Jane ann Underwood of
the parish of St. Bridget's Dublin
28th Feby 1804