I think the warden is the best class to run bow/bow build on. its perfectly viable now, you will be able to solo all over land content and world bosses, and it is possible to even have enough DPS to do dungeons and normal trials.. with the correct builds in place. ( i beleive i've seen some builds of bow bow doing vet trials to or at leased getting over 50k on a dummy )
im not 100% on what the best race would be, i think the redguard restore stamina passive will proc on bow light attacks and single target abilities so maybe redguard will be better for sustain? but khajiit will have a better crit rating so may give more dps? both good choices either way.

Try this: https://forums.elderscrollsonline.com/en/discussion/362978/noss-mirakuru-bowbow-stamina-warden-wrathstone-dlc-ready/p1

Bow bow warden is absolutely solid in both pve and pve. pvp is a bit harder due to how kiting, gap closers and snares are handled in eso making ranged kiting builds harder to play but still strong.

Just understand, that most will insult you if they have one of your snipe casts on their recap screen. Ignore these people, but also remember for pvp, there is much more than hanging in back and snipe spamming. As a warden, get into mid range, control crowds with arrow spray, and setup up your shalks to line up with snipe hits, draining shot helps with this as well.

One of my IC farming / bg alts is a bosmer bow / bow warden with marksman + morag tong + 1kragh, 1 kena or domi or 2pc troll king. Massive sustain and easy leathal with plenty of roll dodge budget.

For pve, either race you mentioned is fine. Just make sure to get your hands on lightweight trap from fighters guild and caltrops from the alliance skill line, both will come in handy at some point but not always mandatory. Only uneducated individuals would have a problem with a pure bow dps in their group.