When to stain and which product?

Also, I live in Tennessee. I believe the new wood is pine but not sure about the old wood. Full sun. There's some green on the railings so I'm assuming that's algae? No problems with prior stain (Armstrong Clark semi-solid).

After you strip and brighten the stained wood and then clean and brighten the newer wood, you can use the TWP Semi-solid colors. The algea will come off with the prep.

Since there isn't any stain on the flipped boards, do I still need to strip them? Also, do I have to strip the old stained railings before staining with TWP or can I just clean and brighten them?

There is a little bit of stain on the old deck floor boards but not a lot. Here's a pic..

Strip the turned boards and any wood that has a previous stain. 

One last question, is there a TWP semi-solid color that closely matches Armstrong Clark semi-solid Sequoia? 

Not really. Might want to test some samples.

https://www.twpstain.com/stain-samples

Ok thanks!

Does it matter if I accidentally get the stripping product on the new treated boards? If it does, would it be better to wait 6 months and just clean/brighten the entire deck since I would be using the same stain that I used previously?

Or should I be more concerned about protecting my deck from the full sun it will receive this summer?

It will be fine if you get the stripper on the new boards but make sure to brighten them after.

Alright thanks again for answering all of my questions!

You can't get new wood to match older wood with deck stains that are transparent or semi-transparent. You would have to test samples on both to get an idea as to the difference in color.

I'm planning on using semi-solid stain--either Armstrong Clark or TWP. Will the new and old boards match better with semi-solid stain?