There are 4000ish people on this list so expect 4000 different oil choices, it is just one of those things.
The brand is not really important however the viscosity is and that will depend upon your temperature range.
And the temperature range is the operating temperature, not the storeage temperature so if it snows where you live but you have a separate blower & the mower just sits down the back of the shed you don't need to put 5 W 30 in your mower.
If you mow in temperatures of 90 deg or hotter ( you have my sympathies ) regularly then you will need an oil that goes up to 40 e.g. 10W 40.
Otherwise plain strait 30 W will normally do.
However you have your plan upside down.
unless you are going to split the engine to change the oil you need to have all the sediment up into the oil so you can get rid of it.
So the best thing to do is to mow your lawn then change the oil when you have finished mowing.
I try to get my customers into the patten of changing the oil as soon as they stop mowing so when you get down to 1 mow a month, change it then so tther will be nice clean oil sitting in there over the off season and they will not be anything in there to settle out.
As for filter & for that matter oil change intervals, oil every season and a filter every second season will usually do a residential mower fine, or put another way oil on 50 fiter on 100 hours.