A Levi marries a female convert. Does the son inherit Levi status? Source?
The halacha explicitly forbids a kohen from marrying a giyores, however, it does not speak about a Levi or Yisrael. Thus, the child of a (male) Levi and a (female) convert is a Levi with no restrictions.
A kohen may not marry a ge’rusha (divorcee), chalalah (woman of defective kohen status), zonah (woman who previously violated certain sexual prohibitions), giyoret (convert) or chalutzah (a Levirate widow). If he does marry any of them, their children likewise become chalalim. Sons born do not have priestly status, and daughter may not marry kohanim.
The Tur, Even Haezer 8 says
... convert or a freed slave who marries a Levite or Israelite woman or a chalalah, then the child follows the status of the man – it makes no difference whether the convert marries the Israelite woman or the Israelite man marries the convert.
Shulchan Aruch EH 4:23 and EH 8:3 state:
גר שנשא בת ישראל, או ישראל שנשא גיורת, הולד ישראל לכל דבר ואסור בממזרת.
A convert who marries a [born] Jewess or a [born] Jew who marries a convert, the child is an Israelite for all matters and may not marry a mamzer.
Mishna in Kiddushin 66b
כל מקום שיש קידושין ואין עבירה הולד הולך אחר הזכר ואיזה זו זו כהנת לויה וישראלית נשאו לכהן לוי וישראל
There is a principle with regard to the hala-khot of lineage: Any case where there is betrothal, i.e., where the betrothal takes effect, and the marriage involves no transgression by Torah law, the lineage of the offspring follows the male, his father.
In the list of the mishna the marriage between a Levi and a convert is not written. The Gemara asks about calling this Kol Makom (This a general principle) is it really a generality (true for every case for which the rule is applied)!? and objects that there are exceptions that are (convert with mamzeret, Chalal with Israelite). The Gemara explains that some cases are skipped in the list because the list of men contains Cohen and illustrates this by the case of Chalala, not written in the Mishna despite than she can marry each of the list of men excepted the Cohen, because Cohen is written. I deduced obviously from this that as a standard case of Kiddushin without Avera, the son of a Levi with a convert is a regular Levi. This case is not written in the mishna because the list of women written there contains only women who can marry a Cohen.