A sectional rodding machine is the best bet.
A basement standpipe is an open-ended and watertight pipe inserted in a floor drain to prevent a flooded basement.
your kitchen drain and floor drain are on same line, and needs to be snaked
If you have a basement floor drain, chances are that you have a clog downstream somewhere, and it's backing up into your basement.
its not easy but if you can locate your main drain pipe and its below your basement floor then saw cut up the floor and attach the floor drain to the trap and then tie on to the main line using the same pipe or adapters. If you have no main drain under your basement floor then you will need to saw cut a hole for a sump pump pit or basin and install the sump pump to your main drain line which has to be installed so your sump pump line runs higher then your main line before wyeing to the main line. then cut up the floor to where u want your floor drain and run the drain to your sump pump basin . Some states want a vent on your floor drain and on your sump basin so I would check what your local codes are.
drain or gully may have another inlet coming into same pipe via a junction under the floor
You have to cut out a section of the floor so there is room for the drain. Something like a foot wide and 16 inches long.
Enough to fill the traps crown weir to prevent sewer gases /fumes from entering the structure
This could be an old floor drain. Check to see if you have another floor drain in a different part of the basement. ...It's not a drain. That is a rough-in or prepared area to install a bathtub or shower. It is normanlly a squared off hole near some other PVC pipe stubs sticking up for toilet and sink connections.
You would have to build a platform for the tub to sit on. About 3 inches if you are using a floor drain. There is a trap right after the floor drain so you would not need one at the tub. If possible you could position the tub directly over the floor drain but that would eliminate the floor drain which would be a problem if the basement every flooded.
Yes, it is done all the time
Without knowing the layout or type of drain pipe and location, it is impossible to give a good answer. Drill a hole in the floor and run the drain with PVC. Drains should fall 1/8 to 1/4 of an inch for every foot of distance. Horizontal distance. There may be a drain line that is closer to the first floor location that you can tap into. Washer drain does not need a vent, it gets it's venting around the washer hose.
Depending on your city or county you can install an above floor basement waterproofing system and drain the water to a drain instead of a sump pump. Waterproof.com has an above floor basement waterproofing system that you can install yourself for around $4 per foot. I hope this helps, if not please elaborate