If I would like to divorce my abusive husband but neither of us are US citizens, can I file a petition for divorce and have him served and then leave the US or do I have to remain here until the divorce is finalized?

You can file here because you live here... however, if most of your assets are in Scotland, you may want to with an attorney abroad before you do so.  Here are are list of things you want to compare to Texas law:
1.  Texas is a fair and equitable state--- this means the courts will usually strive to make a fifty-fifty property division.  The courts can award a bit more if you are an abuse victim and cannot take care of yourself... but this is completely up to the discretion of the judge. 
2.  If your husband has been convicted of assaulting you, then you may be able to ask for marital support.  However, this is again limited to the discretion of the judge.  It is not automatic or guaranteed.
3.  Because your property is located in Scotland, many judges do not think they have jurisdiction over the division of property outside the U.S.... which means you would have to file another suit in Scotland if your local judge declined jurisdiction over the property items. (they could grant the divorce--just not divide the property)
Many people think the laws in Texas are very "black and white" and clear cut.  However, most of the law are subject to "abuse of discretion."  This means that the judge has discretion to do certain things... and as long as the judge is not completely out of line... the appeal courts will rubber stamp his/her decision.  Judges are elected via local politics in every county and have an amazing variety of opinions on the various code statutes.  With that in mind, you also want to set up a meeting with a local attorney so that you get a better feel of the attitudes of your local judge or judges. 
You can then compare those thoughts and ideas with those opinions from an attorney abroad.  You want to know what is going to be the safest and best financial option for you. 
If you don't want to file for a divorce just yet, but you need to be safe, then you can reach out to the locate district attorney's office and seek a protective order.  Private attorney's can also file for protective orders-- but they are free via a district attorney's office. 
As far as where to live... most courts will not require the parties to live in a particular location while a divorce is pending... they only require them to personally appear in court for hearings.  However, if you have children, then the courts will often require the children to live in the county or close by while the divorce is pending.