Disclaimer: I am by no means an expert, and am actually still learning Japanese myself, but I have been interested in these subjects for quite a long time myself, so these are my own observations (I welcome any corrections by more knowledgeable folks):
When polite speech (です/ます/etc) is used and when it's not can be a very complex and subtle matter. New learners are often taught that it is about social standing (senpai/kouhai, employer/employee, etc), and quite often it is, but this is really only one of a multitude of factors that enter into things in real-world interactions. Some of the other factors that often come into play are:
- Social and emotional distance
Is this someone you've just met, or have you known them since childhood? Are they always friendly towards you, or are they usually cold and aloof? Are they somebody you really like, or just somebody you talk to from time to time?
- Confidence / Assertiveness
Plain speech can sometimes express self-confidence or assertiveness (but in other contexts, this can come across as coarse, or aggressive). On the other hand, polite speech (in circumstances where it might not usually be used) can sometimes signal deference or timidity.
- Gender (and gender expression)
Polite speech is often seen as being more "feminine", and plain speech is often more "masculine". Sometimes this is used deliberately, for example, for women to project a more masculine image, or men to seem more effeminate.
Young children pretty much exclusively use plain speech. As they become older, they tend to develop more of a sense of the intricacies of social relationships, and learn to use polite speech, but even once they understand polite speech thoroughly, the use of casual speech is still quite common amongst most young adults, partly because of the people they tend to interact with (largely peers), and I suspect to some extent as a way of establishing their own independence, etc. Older folks (no doubt conditioned by years of workplace interactions, etc) often end up speaking more politely. (These are, of course, all very rough generalizations, and there are many many exceptions.)
Likewise, the relative age between the speaker and the listener can be a factor. In general, the more difference there is in age (either direction), the more likely people will tend toward polite speech. (Some of this comes out of the "social distance" aspect as well. Age differences inherently add some distance.)
Some people who are less confident in social situations may default more toward polite speech (as erring on the side of politeness is less likely to be taken badly, etc). For some, this can even be an unconscious thing where their speech patterns may change depending on their level of anxiety at any given time. For others, talking more casually may be something reserved only for when they are truly comfortable with other people (and can be a sign of real trust, etc).
- Personal feelings towards the other person
The use of unusual polite or casual speech can sometimes signal that you are angry with someone, etc. (For example, if a husband or wife suddenly starts talking politely to their spouse when they don't usually, that might indicate annoyance, or perhaps more serious relationship issues.) This can also be used jokingly, of course.
- Speaker's personality and background
Some people just tend to speak more politely or more casually than others, as a matter of their own personality and the way they present themselves to the world. This may be a deliberate choice, or it may just be a factor of how they were raised, etc. Sometimes, people who speak politely all the time can come off as "stiff" or "off-putting", but sometimes it can be seen as endearing, etc, too. Those who tend to use casual speech a lot can project an image of being "rough and tumble" or "keeping it real", etc. (but sometimes also just "coarse" or "rude")
The particular situation one is in can, of course, play a factor, but an important thing to note here is that quite often, the current context or situation (e.g. being on a date) actually does not have as much to do with this as one might think. In many cultures, the way one interacts with others is a question of what role they are playing at the current time, and changes depending on the setting, but in Japanese, the way one interacts with another has much more to do with who the two people are and their relationship as people, which does not actually change just by going to a different location or doing a different activity.
So, given all this, why might someone be speaking politely even when on a date? There are all kinds of possible reasons (maybe even more than one at the same time):
- At a basic level, he doesn't see their relationship as having changed significantly, just because they're on a date. (Particularly if this is a first date, etc, they're still primarily coworkers first, not suddenly friends or lovers just because of one date, etc.)
- He wants to express that he respects her (not necessarily just as a boss, but also as a potential love interest).
- He doesn't want to seem "too familiar" (i.e. creepy)
- He's anxious about the current situation and can't relax enough to feel comfortable using more casual speech.
- He actually isn't that interested in her romantically and doesn't want her to get the wrong idea (that speaking casually to her might indicate a desire to be closer).
- He might have some emotional repression issues, etc, and using familiar speech in such a situation might feel like more of an emotional/personal expression than he's comfortable with.
- It may just be habit, and habits are sometimes hard to break.
- That's just the way he always talks, or the way he was brought up, and there's really no special meaning to be read into it.
It's also worth noting that the whole question of "are we close enough for it to be ok to change the way we talk to each other?" is a very common problem/trope/etc in personal/romantic relationships (and in particular comes up all the time in anime/manga/etc). There are no hard rules, and it can be very confusing even for a lot of Japanese people to figure out, particularly if they're in the middle of the relationship themselves, so quite often every individual has a slightly different answer.
It's also probably worth noting that in many cases, for those who are fluent in the language and social interactions, the choice of polite or casual speech isn't even necessarily a conscious thing a lot of the time; it's just automatic, based on what "feels" right. Sometimes people may not even be able to explain why they use one or the other in a particular situation without stopping and pondering it themselves for a while first. So you can't necessarily assume that the type of speech being used in a particular instance was actually a deliberate choice, or always has a really logical reason.
As for 「はいです」 and such, the truth is that there are a number of common Japanese expressions that on their surface seem to be polite speech, but in actuality are only used in casual or intimate contexts. This is often because the casual equivalent (「はい」) became so common over time that it ended up being the normal way to say things even in polite speech, so then the more unusual form took on a more "special" meaning, just by nature of its being something people don't normally say. In some ways this is similar to how seemingly neutral words like 「あなた」 ("you") can also end up having other meanings (e.g. "honey"/"dear") in more casual/intimate contexts.
Unfortunately, these are usually just expressions you need to learn are special on a case-by-case basis..