No, neither serving them on one laptop, nor accessing them on one. Boy, that would be a waste if one needs one distinctive computer for each software.
The time is also irrelevant.
Depends on the internet browser: by default just two different windows of the same software should share the same sessions (speak: it should not be possible). But if both windows are using a "private mode" then they'll isolate sessions.
When sitting at work then most likely all people accessing your board from there have the same IP address - session distinction isn't done by the IP address alone.
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