⏰How long do their relationships tend to last?
Koschei is the definition of ride or die. He never gets over a relationship, platonic or romantic.  Never. He carries it for eternity, for better or worse, often to his own detriment.  He is a lot of awful things, but a cheater and a quitter are not among them.  
 
👫Do they display affection in public? What about in private?
Very much, obnoxiously, and proprietarily.  In private, if he truly loves you, there’s little alone time and you might have to actually ask him to ease up and give you breathing space.  If you do it right, he won’t mind, and will happily go off to entertain himself for hours.  He just needs a reminder of boundaries. 

💝How long until they feel secure and comfortable in a relationship? 
It takes a lot of time and emotional yo-yo’ing. The Master is way more cautious than his bombastic disposition would suggest. 

🤐Would they ever confess their feelings first?
HAAAA that’s a great question. Ordinarily, no, and that stupid stubbornness is part of why the Master and the Doctor are still, in canon, dancing in circles around their obvious intense attraction.  Once the object of his affections admits feelings, he will IMMEDIATELY, reckessly reciprocate.  You see this in canon twice with the Doctor (in “End of Time,” when the Doctor says to the Master, “Wonder what I’d be without you,” and in “The Doctor Falls,” when the Doctor says to Missy, “Stand with me, it’s all I’ve ever wanted”).  HOWEVER there have been separate instances in canon (both with Missy) describing the Doctor as her “friend,” a word which means far more than lover in the context in which it’s used.  In my rp’s, it’s been the Master who’s interestingly broken down first and confessed love to the Doctor, and to Jack Harkness, the confession was literally simultaneous.  It’s also complicated by the fact that the Master shows love more often through actions than words.  Putting himself in harm’s way for a loved one, or making himself vulnerable to a loved one (for instance, falling asleep on Jack Harkness’s chest) is an enormous tribute to that person.  

🚀 How far are they willing to go for the person they love?
Already answered here. :3 

🎶Do they have a type? 

The Doctor LOL no I mean yes actually lol, but if you step back and examine the Doctor’s individual traits, which other lovers and spouses he has in other verses share:  he is actually drawn toward compassion, even if he’s the first to deride that personal characteristic as foolish and contemptible.  He’s drawn toward intelligence (you MUST be intelligent, and HIGHLY so), resourcefulness, power, charisma, a certain edge of darkness, humor, and something perverse and iconoclastic, rebellious.  

🏷️Do they give their partners cute nicknames?

Incessantly, yes.  But he also does that to people he dislikes.  It’s just that the nicknames for romantic partners and spouses are gentler and sweeter, if often ornery.  He’s also prone to calling the people he loves names that are affectionate and he absolutely doesn’t mean ( “fool,” “idiot,” “egg,” lol).  

💚Are they prone to jealousy?

HAHAHAHAHA. HAHAHAHAHAHAHAHAH!!!!!!!  H͕̳̮̳͍͈A̜̻̱̦̰͉̗H͍͖͙̗͎͔̟A͉͈͖ͅH̖A̟̣̼̝H̠̮̠͍̘A̛͖͇̠̫̲̮̣H̷͔̝͙̫A̢͈H̹͍̫͇͝A̦̖͓̜̟͚H͕͔̺͚̻̰A̞̞͖̤̝̲͝ḨA͎̱̰̝͡H̯͖̝̲̙A̬̗ ̖̳̗!!!!!!!  Ha. 
 
🐶Are they a cuddler?

Only once he’s very well acquainted, then he’s incredibly physically demonstrative, handsy and clingy.  He feels entirely entitled to lay all over you and expects you’ll be delighted. Peak cat. 

🚀 How far are they willing to go for the person they love? 

To a terrifying extent. There is no moral event horizon; he will do literally anything, except harm his children with the person in question.  Not one other thing is off limits.  He is still learning how to not be a living weapon, and he believes that the fanatical extent of his devotion is “proof” of the veracity of his love.