Yes, travel to Masjid Aisha (Taneem) to enter الإحرام for العمرة after completing all الحج obligations — this has a basis in the السنة from the hadith of Aisha.
Yes, performing an additional العمرة after completing الحج is جائز and has a basis in the السنة. النبي (صلى الله عليه وسلم) instructed Aisha (رضي الله عن her) to go to Taneem with her brother Abdur-Rahman to enter الإحرام for العمرة after she completed her الحج, as narrated in Sahih Bukhari and Muslim.
To perform this العمرة, you must first complete all your الحج obligations, including الطواف al-Ifadah, السعي, and the stoning of the الجمرات. Then travel to the nearest الميقات point outside the Haram boundary — Masjid Aisha at Taneem is the closest (approximately 7 km, easily reached by taxi for about 20-30 SAR). Enter الإحرام there with the intention for العمرة, then return to المسجد الحرام for الطواف and السعي. Some scholars consider this extra العمرة merely جائز while others view it as مستحب. It is best done before الطواف al-Wida (farewell الطواف) if you plan to perform one.
Source: صحيح البخاري, Kitab al-العمرة; صحيح مسلم, Kitab al-الحج