AAIA
I can't answer you question regarding the 1st year FO take home pay but in response to your question regarding 1 night/2night stops rotation.
All rosters commence at LHR as that is your base. The Vegas as an example operates from LGW so on the roster the duty would commence from LHR and 1hr taxi time is allowed for transport to LGW (Company transport is available in the form of a bus but the times aren't always compatible with report times).
In reality most pilots self position to LGW for the report time 90 mins before departure.
You would arrive in Vegas that day 4pm ish.. local. Have 2 nights there (approx 48hrs down route) and then fly home on the 3rd. So a trip beginning on Monday a.m will have you home on Thursday a.m.
A 1 nighter would start for example on the Monday (then approx 24 hrs down route) and you would be home on Wednesday... or more realistically start on Friday and home on Sunday but you get the drift.
For West coast trips the minimum days off will be 3 and the East coast trips the minimum days off is 2. Days off per month vary but I have only had the minimum of 10 days off perhaps twice in 7 years. Usually I average around 12 - 14 but on occasions 20 without leave. It can be a bit of a lottery.
Everyone will handle the fatigue issue differently and it can be tough particularly if you have a long commute. Generally though I am unbearable the day I get home but can be approached by mid morning the following day.
Hope that helps.