Is this a typical practice? It seems like a bad idea to me personally.
It is typical with respect to filing not only the complaint, but also [defendant's] responsive pleadings, responses to a party's request for admissions, claim of appeal, appellate briefs, and various other documents. Although lawful if within the SOL, a plaintiff's unnecessary/negligent delay in filing the complaint pleadings contradicts his interest in an expedited administration of justice with regard to his claims.
Can we submit a corrected complaint and still have it be accepted even though time has run out?
You did not specify your jurisdiction, and it is unclear what you mean by "the complaint was rejected", when that occurred, and why. That being said, in jurisdictions of the US it is possible to file a complaint after the statute of limitations has expired. In fact, the expiration of the SOL is frequently litigated in US courts.
If the SOL has expired, the defendant is most likely to invoke the doctrine of laches and/or the expiration of the SOL among, or in lieu of, other affirmative defenses. In some types of disputes, the court may sua sponte assess the SOL even if the defendant did not raise that defense. See Jackson v. Secretary for Dept. of Corrections, 292 F.3d 1347, 1349 (2002):
even though the statute of limitations is an affirmative defense, the
district court may review sua sponte the timeliness of the section
2254 petition. [...] the district court possessed the discretion to
raise sua sponte the timeliness issue.
By contrast, Wolverine Mut. Ins. Co. v. Oliver, 933 N.E.2d 568, 571 (2001) cites precedents from other jurisdictions concluding that "trial courts may not sua sponte inject the defense of the statute of limitations where the defendant has not pleaded or argued it", although the opinion also cited a case in the sense that "the trial court did not abuse its discretion in raising sua sponte the statute-of-limitations defense for a pro se defendant in order to bar the entry of default judgment in favor of the plaintiff".
If the complaint is rejected upon being entered, the plaintiff will need to establish the tolling of the SOL in order to avoid the dismissal of the complaint.