Staining a deck is DIY 101 and anyone can do it. Spraying and rolling will be more even but in no way will it apply enough stain.
Plus with a sprayer your going to have over spray all over the place, including the siding on the house.
What I have done to speed things up was to use a 1/2 nap roller and as I went along brushed in any missed spots or where it was to thick. Try to do a few boards at a time so you can reach the areas that need to be brush out.
Your working with real wood, some parts are dencer then other's so some areas will just be darker then others. Pine is the hardest wood to get to come out even. Once it's dry if some spots look to light it's because the wood soaked it up, just apply more stain in that area.
The ends of the boards need more coats, that's where the waters getting sucked into the board.
I never would have used water based stain only oil based.
Never use Thompson brand, consumers report rates Behr and Cabits brand way above them, Sikens is also a good brand but very expencive.
The stains that last the least time in order are Clear, Semitransparent, then soild which last the longest.
You will need two coats to get a really long lasting finish.
Tape off the siding with 2" wide painter tape if the sidings up close to the deck boards, if it gets on the siding it's not coming off. I'd use a brush not a roller where it's close to the wall.
Those railings look pretty poopy I'd sand the top 2 X's with a random orbital sander and 60 grit paper before staining.
Do the railings first, before the deck. I use a 4" hot dog roller and tip off with a brush to fill in any missed spots.
You first have to kill all the mold, if not it's going to come right back and come through the stain. A pressure washer with the wash nozzle (lower pressure) and a bottle of bleach connected to the suction hose will work best, Once it's sat for about an hour, rince it all off, let dry a few days.
The under side does not really have to be stained, (will not hurt anything just a lot of time and money for little gain) but I would hit it with the bleach and water to kill that mold.
If you want those top rails to last longer and look far better what I have done is add composit decking on top of it using trim head stainless steel 3" screws. Once there in just tap the holes with a hammer and lightly brush the area following the grain with a wire bruch and most of the screw hole disappers. Now you have a place to sit a drink and it acts like an umbrella to the reat of the railing.
Only use the newer PVC coated composit deck boards so it will not mold up.
Also someone cut the pickets to short on your stairs hand rails, there just barely attached.