This is precisely where newbie questions belong, so no forgiveness needed!
Time stretch basically stretches an item over a longer (or shorter) period of time, so if you stretch the layer it will play the SAME thing over a longer period. This is most obvious with a video file… a 3 second piece of footage, when stretched to 6 seconds, basically plays at half the speed.
I can’t entirely work it out by reading your post, but is the static logo on it’s own layer? If the animation ends up just sitting on a still image of the logo, you can simply drag the end point of the logo layer out, and it will stay on screen for longer. Using time stretch on this is possible, but will have the same effect. I usually access time stretch from the Timeline – if you right click on one of the column headings (such as Layer Name) in the timeline, you can choose which columns you want to display, and if you select Stretch, it will show up with each layer. It starts at 100% and as you drag the number up or down, you will see the layer extend or squash in the timeline.
Otherwise, you could look into the Time Remapping function. You should look at the manual for all the details as it can be a bit of a head-muddle when you first look at it, but it’s very handy for retiming things. In your case, you could enable time remapping, set a keyframe at the point where you want things to stop/hold, and then delete the keyframe at the END. This basically sets a Freezeframe from that point, so that the rest of the layer animation or footage doesn’t play.
Maybe too much info for you here, but those are some options, and good to learn about if you are planning to become adept at AE.