Hi,
This sounds like two opposite things to me. Do you wan't to be off beat or not?
Do you mean quantize to correct your playing, if you hit a key wrong?
I can try to answer your questions this way:
Both the ARP and the SEQ only react to your initial keypress to start and stop. That included recorded midi notes.
The ARP will not start over, if you play legato. The sequencer will start over at each keypress, unless you set the playmode to LEGATO and play legato.
If your initial keypress is off beat, then both the ARP and SEQ will play off beat, unless you have something that quantize your playing.
If you wan't your playing to be quantized, then you can do that in example by using a DAWs input quantize functionality, if it has one. Or perhaps you have other tools to do so. Also in any setup you use.
The ARP and SEQ will allways be tempo synced no matter it's on beat or not, if you have set it to be tempo synced, - provided your sync signal works.