Thank-you for your question. I am a practising immigration barrister with over 10 years experience in UK immigration law.
I understand that you will be applying for an extension of stay as a Tier 1 (PSW) Migrant on 15 November 2010. As to your immigration history, in my opinion the gap in your leave to date will not affect your application. Since you will be applying under the Points Based System, you will have to pass a points-based assessment. You will be awarded points for attributes (which are a United Kingdom qualification, study at a United Kingdom institution, your immigration status during United Kingdom study and/or research, and the date of award of the qualification), English language and maintenance funds. Provided that you score enough points then you will be granted leave to remain as a Tier 1 (PSW) Migrant. The gap in your leave is not relevant to the points-based assessment. It is open to the UKBA to refuse PBS applications for an extension of stay where any of the general grounds for refusal of leave to remain in rule 322 of the Immigration Rules apply, however having a 1 month gap in leave is not one of the grounds. In fact, it is even possible to make a PBS application at a time when you have no existing leave to remain in the UK. This follows from the requirement that "The applicant must have, or have last been granted, entry clearance, leave to enter or leave to remain" in one of various categories, of which Student is one. If it is possible to apply for a PSW visa at a time when you have no existing leave in the UK then a previous gap in leave of 1 month cannot prevent you from making a successful application.
As to the second part of your question, the fact that you finished your course early will not affect your application. Applications for leave to remain as a Tier 1 (Post-Study Work) Migrant must be made within 12 months of obtaining the relevant qualification but if you apply on November 15 then you will satisfy this requirement.
When you apply for the visa you only need to answer the questions that are on the Tier 1 (PSW) application form and provide the information that you are asked for. If you have not done so already I suggest that you have a read through the current PSW application form and policy guidance. These are available here:
I hope I have answered your questions. Please kindly click the green ACCEPT button so that I may be compensated for my time. Thank-you and good luck with your application.