1) Does Logic have a simple way to move a region to the transport bar?
The transport bar? Which position would that be? :-) If you mean the playhead, the answer is yes: Look for a key command called "•Pickup Clock (Move Event to Playhead Position)"
2) Is there a way to trim left/right of the transport bar on a selected region?
You probably mean left/right locators here, right? Maybe what you're looking for is on this list of locator related key commands:
Set Locators by Regions/Events/Marquee
Set Rounded Locators by Regions/Events
•Set Locators and Play
•Set Rounded Locators and Play
•Set Rounded Locators and Cycle Play
•Set Rounded Locators and Record
•Set Rounded Locators and Cycle Record
Set Punch Locators by Regions/Events/Marquee
Set Rounded Punch Locators by Regions/Events
Set Punch Locators by Cycle Locators
Move Locators Forward by Cycle Length
Move Locators Backwards by Cycle Length
Toggle Auto Set Locators
Set Locators by Marker and Enable Cycle
Set Locators by Previous Marker and Enable Cycle
Set Locators by Next Marker and Enable Cycle
Zoom to fit Locators, store Navigation Snapshot
Select Inside Locators
Deselect Outside Locators
Delete inside Locators
Delete outside Locators
Split Regions/Events by Locators or Marquee Selection
Snip: Cut Section Between Locators (Selection)
Snip: Cut Section Between Locators (Global)
Insert Silence Between Locators (Selection)
Insert Silence Between Locators (Global)
Repeat Section Between Locators (Selection)
Repeat Section Between Locators (Global)
Adjust Tempo using Region Length and Locators
Time Stretch Region Length to Locators
Trim Regions to Fill within Locators
Crop Regions outside Locators or Marquee Selection
What exactly do you want to do? Set the left/right locators at the same position of the region start/end? This can be done automatically, every time you click on a region, if Auto-Set Locators is enabled. Control click in the bar rules, and you'll see a long list of options, including a way to enable Auto-Set Locators. You can also use the first key command on thelist above if you just want to set the locators according to the regions start/end points once.
If you mean the opposite - to crop the region at the left right locator positions, just select the region and press Crop - either as a key command (Control C) or by using an icon in the tool bar.
I actually assume "Crop" is what you want. OTOH, if you want to divide the region at the left/right locator positions, there are several ways to do that as well. Try to select the region and a long Command-click inside the locators in the bar ruler,. Or press press ⌘\. Or, Control-click on the region, and look inside the Contextual menu (look for "Split"). Or use the Region menu (in the Split submenu).