TNG, post season 2, was largely sexless as well since the "evolved humanity" Gene wanted... Geordi wasn't running around telling how oh-so-naughty Corsair was (for one of many examples). TNG didn't waste week after week on hormones because it wasn't an oversexed soap opera either.
I have no idea how you can claim a show with William T. Riker front and center is in any way sexless... ;-)
As a shameless flirt and a bi dude, I really never minded Jack. As for River, her
immaturity is sort of a consequence of her rather atemporal upbringing, more-or-less imprinting on a man from birth, which would fuck anyone up.
As for the Doctor's sexuality and activity... well, it's somewhat strange, mostly unspoken:
Clearly they have had children, which assuming the looms are bunk, would necessitate some... intimate liaisons. Certainly Missy implies there's more to their relationship than just friendship as well.
One does get "engaged" to Cameca in
The Aztecs, but seems equal parts interested and "little kid who doesn't know what this is all about" about it. But definitely not sexual.
Slightly touchy subject, but Two and Jamie have certainly been interpreted by some fans as
more than friends, what with all the hugging, though certainly extra-canonically.
Three is the first Doctor to show any sort of twinge of romantic feelings when he becomes jealous of Liz's beaux
[EDIT: Jo's beau]. Granted, that happens over the course of the single episode and Liz's
[EDIT: Jo's] last appearance, but the ending of it the Doctor is very clearly lamenting the "one that got away."
Four is usually considered to be the one with relationships, there's the will they/won't they feeling to him and Sarah Jane, outright "opposites attract" flirtation with Romana I, heavily implied relationship with Romana II and basically a pseudo "Father/Mother/Son" relationship with Romana II & Adric.
Five seems to be disinterested and has a more sibling-esque relationship with his companions. However, Nyssa kinda seems to imply emotional attachment with the Doctor beyond friendship in her final episodes. Big Finish has played this up a little bit as a one-sided crush in the audios where they travel alone together.
Six and Seven are more or less asexual beings across the media, with the exception of Seven in the context of the "Human Nature" novel.
Eight is far more emotional and the first Doctor to smooch on camera. His first novel appearance, in his handsome new body, has him sleep with Benny. (Under the assumption that she always liked the Doctor, but his new appearance made the deal which speaks ill of a rather complex character like Benny.) His audio appearances have him more in the asexual vein, as his feelings for Charley are
[EDIT: largely] left unspoken and Charley is basically unrequited.
Nine and Ten are active romancers and kissing and I would say it's silly to assume Ten and Rose never had sex. Eleven returns to the asexual mold, though he tells a lot of stories about hookups with priestesses and movie stars and is clearly sleeping with River sometimes. I do not think he and Amy or he and Clara ever happened, though - unfortunately - not for lack of trying on both their parts.
Twelve's sort of odd. Not many opportunities to meet up with a one-off hookup in his stories, clearly not with Clara. We can assume he was with River at least once in this form. Thirteen so far seems pretty low key.