I don't think you're ever required to any answer any questions from the police, whether avoiding self-incrimination or otherwise. You're only required to answer a question in court, and even then only if the answer wouldn't be self-incriminating (or a few other narrow exceptions I think; the concept of the court not being allowed to compel someone to testify against their spouse is a common trope in media, although I'm honestly not certain how accurate it's portrayed). You also aren't required to take the stand when accused of a crime; while you can choose to do so, you're also free to just have your lawyer make your case via the questioning of witnesses instead of having to answer questions directly yourself.
That said, my understanding is that you're _not_ allowed to plead the 5th if the answer wouldn't actually be self-incriminating, so it's a weird thing where you're only allowed to not answer a question by essentially stipulating that you _did_ do something illegal that would be disclosed if you answered truthfully. If they can prove you weren't actually avoiding answering due to self-incrimination but plead the 5th anyways, I'm pretty sure you can be charged with contempt of court. Having never been on a criminal jury, I can't say I know exactly how it would play out in deliberations, but it's hard for me to imagine that it doesn't affect things at all; even if a jury isn't technically allowed to consider it an admission of guilt, from a legal perspective pleading the 5th seems pretty explicitly either a non-legally-admissible admission of guilt or a crime of contempt of court in itself, so I don't see how the law isn't basically forcing the jury to conclude that you've committed a crime one way or another. The question would then boil down to which of the two crimes the jury thought you had committed (the one you were accused of or contempt of court), and while they're not supposed to be deciding the question of the latter, it seems likely that the jury's view will be tainted by this.
Of course, all of this only applies if you did actually commit a crime; if you genuinely didn't commit any crimes, you wouldn't be lying under oath when stating that instead of pleading the 5th. The jury still might think you did commit the crime though and are just doubling down on lying under oath to try to hide that, though.
I don't think you're ever required to any answer any questions from the police, whether avoiding self-incrimination or otherwise. You're only required to answer a question in court, and even then only if the answer wouldn't be self-incriminating (or a few other narrow exceptions I think; the concept of the court not being allowed to compel someone to testify against their spouse is a common trope in media, although I'm honestly not certain how accurate it's portrayed). You also aren't required to take the stand when accused of a crime; while you can choose to do so, you're also free to just have your lawyer make your case via the questioning of witnesses instead of having to answer questions directly yourself.
That said, my understanding is that you're _not_ allowed to plead the 5th if the answer wouldn't actually be self-incriminating, so it's a weird thing where you're only allowed to not answer a question by essentially stipulating that you _did_ do something illegal that would be disclosed if you answered truthfully. If they can prove you weren't actually avoiding answering due to self-incrimination but plead the 5th anyways, I'm pretty sure you can be charged with contempt of court. Having never been on a criminal jury, I can't say I know exactly how it would play out in deliberations, but it's hard for me to imagine that it doesn't affect things at all; even if a jury isn't technically allowed to consider it an admission of guilt, from a legal perspective pleading the 5th seems pretty explicitly either a non-legally-admissible admission of guilt or a crime of contempt of court in itself, so I don't see how the law isn't basically forcing the jury to conclude that you've committed a crime one way or another. The question would then boil down to which of the two crimes the jury thought you had committed (the one you were accused of or contempt of court), and while they're not supposed to be deciding the question of the latter, it seems likely that the jury's view will be tainted by this.
Of course, all of this only applies if you did actually commit a crime; if you genuinely didn't commit any crimes, you wouldn't be lying under oath when stating that instead of pleading the 5th. The jury still might think you did commit the crime though and are just doubling down on lying under oath to try to hide that, though.