Hi josh,
我管理着一个维护和清洁公司,如果grout is good, why not spend a few hours cleaning it 1st, we use neat toilet bleach and bi carbonate of soda and mix it to a runny paste and scrub with a firm extra large toothbrush style brush, put it on, leave it 5mins, scrub and re apply, then leave for 15 mins then wash away.. Then assess what really needs replacing and what doesn't

It would be best to remove all grout as it would be a more consistent coverage and help with longevity .

Yes better to remove all grout, to make same colour nice an good, is not easy to remove old grout,but for quality job your must done.

How wide is the gap, is it coming out easily ? I never came across any grout that would come out easily totally

Of course it’s better to remove the old grout before applying new but removing old grout it’s not a easy job at all
It can be done manually but it will take ages to do a small area but if ou have time on your hands why not good luck

Other option it will be with a power tool but that will unfortunately chip the tiles

Hi would stick with the deep cleaning and small repairs where you can do them.

Best to remove all grout, just incase the colours don't match