In languages which have gender-like classifications for nouns, like French and Russian, how often do nouns change gender over time? Have any studies been done to get statistics on how many words have changed gender in various languages over the past, say, 1000 years?
There is no simple answer.
In languages with gender - less than half* - the gender of specific words often varies not just by time but by dialect. Likewise it varies across languages in a language family, for example the Romance languages. (And specifically for cognates, not just for loanwords and neologisms.) Using the modern Romance languages and Latin as a well-documented example, you can start to build a model for that case.
To that end: in another question here on Linguistics SE, user @LePressentiment referenced The Genders of French and Spanish Noun Cognates: Some Statistics and a List by Teschner, Richard V., Canadian Modern Language Review, v43 n2 p256-66 Jan 1987.
Change For One Noun
Within a major language you can track competing variants over a relatively short time: Note that they often have different senses - arguably distinct words. (In Romance, feminine often implies an abstract or poetic meaning.) This method has too many problems.
Change Across Languages
You can examine a word like dolor, which was masculine in Latin, and in Middle French, but is now feminine in French and Ladino, and both masculine and feminine in Occitan and Catalan, but still masculine in other major Romance languages. We also know the approximate dates of when these languages split from each other and from Latin.
Likewise you can examine Proto-Indo-European dṓm, which was feminine originally and in Latin, but masculine in Slavic, vulgar Latin, Greek and Sanskrit, and neuter in Germanic and none in Armenian, because Armenian and Persian lost the gender feature. See below
Change of the Noun Class Definitions Themselves
One major assumption is that the concepts of gender will still exist in a language in 1000 years. Gender itself is not stable over 1000 years: the Romance languages, English, Dutch and Scandinavian languages all went through some sort of genus shift over the past 1000 years, in some languages still unclear or ongoing: https://en.wikipedia.org/wiki/Vulgar_Latin#Loss_of_neuter https://en.wikipedia.org/wiki/Gender_in_Danish_and_Swedish https://en.wikipedia.org/wiki/Gender_in_Dutch_grammar#Overview https://en.wikipedia.org/wiki/West_Frisian_grammar#Gender
And though we may tend to think of gender, ie two or three noun classes, some (non-Indo-European) languages have more than a dozen classes. In those languages the classes tend to be more regular. In any case there is not always a clear line between a class like gender and a declension pattern, but here is one global count:*
Note that it is often an areal feature, ie potentially unstable upon contact with unrelated languages:
We would not expect a language where the class is obvious given the word to change like one like German where the gender of many words is opaque and necessarily memorised, or Hebrew which was suddenly revived after a thousand years, or English which has very limited grammatical gender beyond pronouns.
A Realistic Model
In the end, we cannot possibly speak of a universal constant rate of change of the gender feature, because there is no universal constant rate of language change in general. Language change depends a great deal on societal upheaval, writing, foreign influence, politics, geography and so on.
So perhaps you could arrive at a formula that - given many parameters - yields an estimated probability of gender stability for different scenarios (eg if literacy is maintained, if something like the internet comes along, if China unseats the US as the global superpower, if the Martians invade...).
* Global counts of language features are problematic because there is no perfect definition of a language (vs dialect) or how to weight them. Should Luxembourgish count the same as Urdu?