No - taking the action out of the stock is fine.
The smaller stud goes in the front but you must ensure that it does not go throught the wood and conflicts with the action/cylinder. You can chop off a bit of the end but it is usually not required to do this and you sacrifice potential grip.
I use about 5 or 6 bits starting off with 2mm and going up in increments. Be very careful as if the hole is too small you risk splitting the wood and if it is too big, the stud will strip and not stay in. Also - once drilled , it stays that size so check twice drill once.
I usually get a point and make a depression in the stock where I am about to drill using a nail or similar. If you can grip the stock in a vice or workmate (suitably protected) then all the better.
For the rear, the only main consideration is not putting the hole to close to the end (as it is for the fore end too) and that the butt plate is usually screwed on so the stud your putting in may strike the screw that is already there. I usually remove this screw, hold it against the bottom of the butt to see the degree of prenetration then give it an extra bit for security.
Take it easy and dont rush. Sharp bits are really essential. I usually just de-burr the finished hole a little too.
If you have a junk peice of hard wood or your not sure if the final hole should be 4mm or 5mm, then take the butt padd off completely and do a test hole to see if one size is too small and another too big, That way you are certain and dont cock it up.
Good luck.