By properly sizing the vent by fixture units and developed lengh and pitching it up from the fixture it serves
You have a blockage in your main sewerline. You have a problem with your venting system. This is why some codes now require a vent within 6" of the toilet. If you will add a vvent to your main line at the toilet you will stop this. Check to make sure your vents are not clogged by a bird or bees nest first.
1/4 inch per foot.
That would depend on the height of the drain and how deep the sink is. As long as the drain is lower than the bottom of the sink, it should be fine.
Because the washing machine drains down the same main drain and it's suction as it goes by is trying to suck the water out of the under sink U trap.
Someone connected a storm line to a saniraty or visa vera
They should not be cleaned in the kitchen sink and YES, it will harden in the drain.
drain line is partially clogged.
Kitchen sink and washer use same drain line. Drain line clogged causing backup of drainage from washer into kitchen sink. Have drain line unclogged to take care of problem. If drain line not clogged, check for proper venting of drain lines.
A vacuum is being created at some point after the two drainage pipes meet. This forces air back up the other pipe and through the "U"-bend underneath the sink, causing the water to gush upwards and drain again, making the gurgling sound. Since both the bathroom and kitchen gray water supplies contain no sewage, this not harmful.
Maybe 600 for a kitchen sink to be installed.
yes.
The overflow is built into the sink, it is not part of the drain. If there is no overflow, you can not add it.