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Since it's a good idea to keep weeds down within raspberry patches, growing monkey grass should help, as this groundcover will eventually fill in empty areas. I do not think it will choke out your raspberry plants. Monkey grass is seldom an invasive problem, provided you have the clumping variety. However, I would make sure to keep the raspberry plants pruned as these can get quite large and unkempt in appearance without annual pruning.

Your best bet would be to hand pull as much as you can and then add mulch to keep weed growth to a minimum. However, you could also try sprinkling corn meal in the area, which has a chemical in it that acts as a pre-emergent on plant seeds preventing them from germinating. Sprinkling corn meal will not harm plants but will keep weeds from growing. Another option (though not one I enjoy recommending) is the use of conventional weed killers. Here is an article that you may find helpful: https://www.gardeningknowhow.com/plant-problems/weeds/conventional-weed-killers.htm

You have a few options:

I would recommend pouring boiling water on the areas where the weeds are growing. Boiling water will kill whatever it comes into contact with.

Alternatively, you could cover liriope clumps with cardboard boxes or use a gallon milk jug with the bottom cut off as a shield. Spray the grass with an herbicide, such as Round-up, during hot weather when the grass is actively growing. One application won't work, as it's going to take several. If a little herbicide touches the liriope, don't despair. It will only kill the little sprout, not the clump.

Ortho Grass-B-Gon will kill grass and weeds but won't harm your liriope.

Monkey grass is a tough plant and will thrive in nearly any type of soil. You could mix in some organic matter and cut the dead growth off so the new growth can come out quicker.

Monkey grass can be easily lifted and divided in spring. Simply use a spade shovel to cut the clumps into the desired pieces and replant elsewhere.

Yes, you can. In fact, cutting it back some during transplanting can help reduce the possibility of shock. These articles will help:
https://www.gardeningknowhow.com/ornamental/groundcover/monkey-grass/pruning-monkey-grass.htm
https://www.gardeningknowhow.com/ornamental/groundcover/monkey-grass/transplant-monkey-grass.htm