Richard III, Globe Theatre, London The Doctor's Dilemma, Lyttelton, National Theatre, London

Mark Rylance plays Richard III as a charming rascal whose dastardly deeds seem little more than amusing pastimes

It's seven years since the former artistic director of the Globe strutted on its stage, but Mark Rylance is back – as Richard III, an obvious heir to Jerusalem's Johnny "Rooster" Byron. Here again is a man who staggers on with a limp... but one that'll never stop him capering.

A hunchbacked Rylance holds close a tiny, withered arm. He speaks with a stammer – but it lends a kind of comic bumble. For this is a jovial Gloucester. Rylance may be facing personal sadness with the recent death of his step-daughter, but he steps on stage with a snigger, gurning at groundlings. Rather than an evil, sinister schemer, this Richard is merely a cheeky rogue.

Even with Rylance's deliberate stumbling, the verse is clear as a bell, and that goes for the rest of the cast too, presumably thanks to the director Tim Carroll's textual work (although he has given the script a good prune).

It's an all-male cast, and female characters move with a creepy gliding motion, as if on castors. In a play about artifice, there is little attempt to render them emotionally true (and Queen Margaret is exorcised altogether). Instead, cross-casting serves to heighten the mannered dissembling of the court; Carroll makes sense of Richard's unlikely wooing of Lady Anne (Johnny Flynn) by emphasising the affected nature of her lament. So how does Richard win her? Rylance becomes still, speaks softly. It's an audacious seduction, but an unflashy one, played as if genuine.

It's not, of course; Richard's simply better than everyone else at putting on an act. Indeed, in a court so full of fools – Samuel Barnett's uppity Queen Elizabeth, Colin Hurley's wheezing, flappy King Edward, Paul Chahidi's dim Hastings – seizing power is like stealing candy from a baby. Rylance plays the mock-faithful servant so playfully that lines such as "I am too childish-foolish for this world" seem newly apt.

There is little scheming between Richard and Buckingham (an underwhelming Roger Lloyd-Pack), and their dastardly deeds seem merely amusing pastimes. The solution to a problem – "chop off his head" – is exclaimed as casually as by another Carroll's Queen of Hearts; later Rylance tosses about Hastings' decapitated bonce as if it were a turnip. Buckingham's speech "persuading" Richard to accept the crown is full farce, complete with monks' habits and audience participation.

Playing it all as madcap escapade makes sense of Richard's most brazen moves. The request for strawberries shortly before decreeing a death is in keeping with Rylance's glib pantomime villain. Carroll has Lady Anne hold Richard's hand even as he announces she is sick and he intends to marry another. This is less chillingly calculating than bloody cheeky.

But it is still, supposedly, a tragedy. And in the final scenes, we do see a man increasingly confused, erratic. Richard's battle-eve speech reveals a glimmer of the small person inside, as a little, low voice confesses – without much drama – "I rather hate myself".

Even if he is losing his grip, Richard staggers into battle a fierce firebrand. But this (rather than a dodgy dream sequence of sheet-draped ghosts) is where Richard finally faces his demons, as he is confronted by those he has slain. It's a neat scene, showing Richard's mental as well as physical degradation; when Richmond finally stabs a sword through that poor hump it feels an ignominious end.

This is a terrifically enjoyable production, more full of laughter and buffoonery than any Richard III you're ever likely to see – well-suited to the Globe, and to Rylance's prodigious talents as charming rascal. But such an interpretation inevitably leads to serious losses – notably, of the nasty menace of his machinations, or any exploration of the darker side of human nature.

Playing for laughs, rather than addressing the cost of playing with human lives, mars The Doctor's Dilemma too. George Bernard Shaw's 1906 play is billed a tragedy but is also closer to farce.

Sir Colenso Ridgeon – a controlled but deeply expressive Aden Gillett – has just been knighted, for finding a cure for tuberculosis. His doctor pals congratulate him, and, while none seems convinced of his discovery, it's all back-slappingly chummy. Each has their preferred – often silly, pseudo-scientific or expensive – medical methods, be that fashionable surgery or "stimulating the phagocytes" (full credit to a wonderfully bumptious Malcolm Sinclair for turning this into an unlikely comic catch-phrase).

But, overall, the joke feels laboured: while the director, Nadia Fall, has trimmed the script, a sharper scalpel could still be taken to Shaw's fleshy text. Fall goes hard after any laughs, but there's a lot of doctor lampooning before we reach the dilemma.

A woman turns up begging Ridgeon to cure her husband, Louis Dubedat, a talented artist. He's roundly declared a genius, but it soon emerges he's a lying, cheating scoundrel who "doesn't believe in morality" – a fact of which his wife, Jennifer, is unaware. Also needing the cure is Ridgeon's old, faithful friend, the honest, bathetic Dr Blenkinsop. Ridgeon can save only one man: which will it be?

Actually, Blenkinsop soon becomes a side note. The urbane debate Ridgeon begins over whether it's better to save a good but talentless man, or a bad one who paints wonderful pictures, is soon muddied by a stronger emotion: love. Or sex, at least. Ridgeon confesses that, were Dubedat to die, he'd be straight after his gorgeous widow...

Genevieve O'Reilly conveys Jennifer's credulity without reducing her to a simpering idiot, and there's a sexy sparkle with her self-absorbed hubby, an engagingly louche Tom Burke. But if Dubedat is grossly selfish, so is Ridgeon. He claims to have Jennifer's best interests at heart, but his own motives are hardly pure.

The Doctor's Dilemma is staged well, but the play isn't one thing or the other. The black cynicism and gallows humour are enjoyable, but the doctors' blasé attitudes, tragi-comic deathbed scenes, and unappealing characters mean that, in both love and death, the stakes don't feel very high. It neither tugs at the heart strings nor fully tickles the funny bone, making positive diagnosis tricky.

'Richard III' to 13 Oct (0207 401 9919); 'The Doctor's Dilemma' to 12 Sep (020 7452 3000)

Critic's Choice

Director Nicholas Hytner shifts Shakespeare's riches-to-rags saga Timon of Athens to our own era of fat-cat bankers and the Occupy movement. Simon Russell Beale shines in the title role, at the NT Olivier (to 31 Oct). Gregory Doran's superb new RSC Julius Caesar, set in modern-day Africa, arrives in London too, at the Noel Coward (8 Aug to 15 Sep).

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