I have a Muslim friend who I would like to invite for dinner at my house. I am not Muslim. I understand the basic halal rules, and would try to follow them. Would she and her family be allowed to eat food prepared in a non-halal kitchen?
No, problem with you cooking in a non-halal kitchen and they eating it. You could be cautious to not bring things that are non-halal into contact with those things that are halal (which as you said you know, so not getting into the details of it) and which you shall be serving. Contact also includes your hand touching pork and then without washing your hand you touching the halal food.
If you're not a Chirstian, Jew or a Zororastrian then make sure that in front of your friend at least you don't touch the food just to make sure the food is ritually clean.
Source: Rulings of Ayatullah Sayyid Ali Husaini Sistani.
Islam prohibits only what is haram...It is immaterial if a halal item is made for, or served to you by a Muslim, Christian, Jew, Hindu or anyone else.
Source: Food prepared by Non-Muslims
No problem, if you know what is haram then avoid it fully, and enjoy the meal Because its mentioned in quran(the haram things are dead meat, blood, flesh of swine, alcohol-ethanol to be specific, and a few more things), irrespective of who made the meal. The translation of the following Quranic verse should clarify your doubt; Surah Maida(5), verse 3
5:3 Prohibited to you are dead animals, blood, the flesh of swine, and that which has been dedicated to other than Allah , and [those animals] killed by strangling or by a violent blow or by a head-long fall or by the goring of horns, and those from which a wild animal has eaten, except what you [are able to] slaughter [before its death], and those which are sacrificed on stone altars, and [prohibited is] that you seek decision through divining arrows. That is grave disobedience. This day those who disbelieve have despaired of [defeating] your religion; so fear them not, but fear Me. This day I have perfected for you your religion and completed My favor upon you and have approved for you Islam as religion. But whoever is forced by severe hunger with no inclination to sin - then indeed, Allah is Forgiving and Merciful.
As said in the above verse, it is clear what is Halal and Haram, irrespective of who cooks or slaughters or cuts the animal or plant.