Read the answers in dark blue.
n 1 1) Can we have the manufacturing date populated in the quant instead of GR date. we are exploring this options to unable the FIFO logic to work fine.
FIFO logic works fine by the default if your physical process is working as it was supposed to. If you still need to have the GR date updated with manuf.date, you need to have a manufacturing date stored somewhere, and then you can use some user exit to overwrite GR date field with this MF date.
2) 2) Can we have the quant created in TR , when we do the C011N. I know that quant gets created when the TO is created.
Quant cannot be created in TR.
3) 3) Does std SAP, allow the GR date getting populated in quant, when moving from one storage location to another.
Why would you want that? It is much better to have the actual initial GR date, you son't want that changed when you do a mvt on the quant (or do you?)... I'm interested in your business scenario for requiring this! 
4) 4) Does std SAP, permits the stock move from one bin to another in LT01 along with the stock type together.
Yes.
1) 5) How to find the menu path in SAP, if you are not sure for a particular T-code in SAP
By business processes, if you are looking for t-code for a logistic processes you will choose that node, then go to more specific processes until you find the exact t-code you are looking for.
6) Does Std SAP allow for moving the material from storage location to another in SAP WM for a same plant. I know, it works fine in IM by using the T-Code like MB1B with movt type 311 and MIGO_TR.
Sure, MB1B - 311 works fine, with only one more step, create(and confirm TO in WM). You can use any process with WM just as you would with IM, it is just like IM extended with WM functionality (TOs).