Safest and most effecient is a steel linner inside chimney and extending prescribed distance above roof.

Most definitely up, last thing you need are sparks going up the wall of your house and smoke into the soffits. I really doubt that a side vent could be legal for a wood or coal stove unless you take it to the required height above the roofline. At which point you might was well use your existing chimney and avoid the ugly appendage. Either use a stainless liner or line with concrete and you should be good to go. Most old flues are too big for modern stoves, so you'll get better performance as well.

Definitely use the chimney with a liner and correct vent cap. Here are the applicable codes. http://www.gov.allconet.org/permits/bulletins2/TB%2021%20-%20Technical%20Bulletin%20Masonry%20Chimneys%20Fireplaces.pdf

I would recommend hiring out for this project considering tuckpointing is an art and requires some experience and you need to make a new opening in the roof, and flash it properly, and follow all codes including height, lateral loads, etc. You might need a roofer and a tuckpointer actually.