Just trying to figure out how Trailing Stop works.

Lets say I have a trade initiated with a TP=24, SL=24 and TS=2.
Now for simplicity sake I'm going to say 0 is my open price so SL would be -24 loss and TP would be +24 profit. Now as the day progresses and the price goes in my favor to +26 TS gets triggered because I'm in extra profit of +2.

Now two questions;
1) Does SL now move up to +26 so just in case the market takes a turn for the worst, I'll still get my +25 pips profit instead of loosing and hitting my original SL?
2) Why didn't TP close my order at +24 in the first place?

Thanks