I believe so. A motion for new trial and any motion to "modify, correct, or reform" a judgment must be filed within 30 days of the date the final judgment (or, in your case, the divorce decree) is signed. Among your examples, a motion for sanctions is a motion which "modifies, corrects or reforms" a judgment and, therefore, must similarly be filed within 30 days after the final judgment is signed. See TRCP 329. Motions for new trial and to modify, correct, or reform a judgment have the effect of extending the trial court's plenary power for up to 105 days from the date the final judgment was signed. However, I do not believe this means that you can, for example, file a motion for new trial within 30 days after the date the final judgment is signed and then a subsequent motion for sanctions 40 days after the date the final judgment is signed. All postjudgment motions must be filed within 30 days after the final judgment is signed.
Those do not look like post-judgment motions to me. Findings of fact and conclusions of law, would be too late, notice of appeal you should be good. You need to hire an attorney and quite messing around with motions that are not proper when a judgment is already entered. You are not going to catch the rabbit down them trails.
All depends on whether the motion for new trial is granted. It's too late for discovery, striking pleadings, etc. unless the new trial is granted. Why don't you talk to a lawyer?
I am not intending this to be legal advice, because I don't know the particulars of your situation. Call me if you would like to discuss this or other isues.