If by objects you mean notes, then yes. All the standard nudge commands work. If you mean graphic objects that you have inserted manually; I'm not sure. And I'm not in front of Logic right now so can't check.
One can also nudge notes up/down in pitch (and even 'nudge' notes to the next/prevoius voice by using key commands to the next polyphonic voice ( = MIDI channel), or even to the next staff.)
Personally I use Alt + arrow uo/down/left/right for nudging pitches/positions, and have set Alt + Left and Alt + Right to nudge "by division". This way, I can change the nudge value simply by changing the division value (default is 1/16) in the Transport.
This means fewer key commands to remember. There are also two useful Icons in the Toolbar, "Nudge Left/Right", and "Nudge Value", which also work for the score editor.
atm I'm working on a clickless project, everything is played 'live' and our click is the drummer and a sort of average of what everyone thinks the tempo is
( it sounds better than that sounds IYKWIM)
so grid increments aren't that useful. Ticks would be great if holding down the key would repeat clicks. I've opted for now for half-frames which seem to give 20 ticks per click @ 120 bpm.
Even with click-recorded material, nuances of phrasing and synching with human players requires a more flexible nudge protocol. I like click-CTRL-drag but unfortunately that takes both hands and often requires zooming in to get better precision - then zooming out again to see the landscape. as I type this I wonder how many LUGgers use 3 monitors?
... and choose whatever increment you want from the Nudge Value list:
Visibility of note shifting in the score depends on the score/region status of Quantize and Interpretation. In case you chose ticks, the event list shows you that every keystroke adds or subtracts 1. This works for position and length. And yes, the key repeats if you hold it down. If your part runs in a cycle, you can fine-tune the timing in the smallest steps available.
I think the Nudge Value setting serves what you require. Instead of chosing the value from the list you can also set the Nudge Value on the fly. For me this is too complicated but the functions are here. Just open the KeyCommands window, type "nudge" into the search field and look at the options.