Swallowing saliva whilst praying does not affect the prayer, because it is not food or drink and it does not come under the heading of food or drink in any way. If a person is fasting, his fast is not invalidated by swallowing his saliva, so it is more apt that the prayer should not be invalidated by that.
If a person who is praying swallows remnants of food that are left between the teeth, that does not invalidate the prayer, if it is something small that flows with the saliva, because it is difficult to avoid that.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: What is the ruling on remnants of food between the teeth whilst praying?
He replied: With regard to remnants of food between the teeth, there is nothing with them remaining between the teeth, even if one prays, but if anything breaks off of it, he should not swallow it. Sometimes some food gets stuck between the teeth, then after a while it comes out from between the teeth or he may dislodge it with his tongue and it comes out. We say: There is nothing wrong with that, but he should not swallow it. End quote.
Al-Bahooti (may Allah have mercy on him) said: There is nothing wrong with swallowing remnants of food that are left in the month, without chewing, or swallowing remnants of food that are left between the teeth, without chewing, that are small and will flow with the saliva, because that is not called food. That which does not flow with the saliva, rather it has substance, invalidates the prayer if it is swallowed… End quote from Kashshaaf al-Qinaa‘, 1/339
It says in al-Mawsoo‘ah al-Fiqhyyah (27/124): The fuqaha’ are unanimously agreed that the prayer is invalidated by eating and drinking in general, but they made an exception with regard to that which is caught between the teeth and is less than the size of a chickpea. This does not invalidate the prayer if it is swallowed, but they stated clearly that the prayer is invalidated by chewing if it is a large amount. End quote.
With regard to the smell and flavour of the toothpaste that remains in the mouth, that does not affect the validity of the prayer, because it is not regarded as food or drink.
And Allah knows best.