I understand that in order to craft something I convert its worth into silver, multiply the dc by your roll, and subtract that number from it's worth in silver to determine your progress. This represents progress made during a week. How does time pass during this week? If I had a party member do this would they be allowed to participate in adventuring until the next week where they'd roll again, or are they completely dedicated to crafting for this period?
Talk to the DM
The limits of what other tasks can be undertaken while crafting mundane items is unstated, therefore dependent on the DM. Guidance for the DM's house rules, however, can be gleaned from the magic items crafting rules here:
The creator also needs a fairly quiet, comfortable, and well-lit place in which to work. Any place suitable for preparing spells is suitable for making items. [...] The caster works for 8 hours each day. He cannot rush the process by working longer each day. But the days need not be consecutive, and the caster can use the rest of his time as he sees fit. A character can work on only one item at a time. If a character starts work on a new item, all materials used and XP spent on the under-construction item are wasted. (DMG 283)
Without corresponding rules for making Craft skill checks, an argument can be made that none of the above is strictly necessary, but most DMs won't go for that. Instead, a house rule might look like this:
The craftsman needs a reasonable and appropriate place to ply his trade. The craftsman works for 8 hours each day. He cannot rush the process by working longer each day. But the days need not be consecutive, and the craftsman can use the rest of his time as he sees fit.
A reasonable and appropriate place will vary based on the Craft skill employed as the difference between where one plies his Craft (blacksmithing) skill and where another his Craft (poetry) skill will be vast. Further, as craftsmen start new tasks without first completing old tasks but can return to incomplete tasks and complete them, the last part of the magic item creation rules appears burdensome.
Even a generous DM will probably prohibit making faster or greater progress on items if the character spends 2 or even 3 8-hour shifts using his Craft skill. The game prefers characters do one thing at a time, rarely rewarding overtime.
The house rules detailed above don't prohibit simultaneously crafting and adventuring, but make adventuring secondary while crafting.