During the WWII, did the German Luftwaffe ever have mixed units of Bf-109s and Me-262s, and if so, which units and/or which airfields?

Would you mix petrol and diesel engines in a factory or a car repair shop? Of course not. Airplanes are vastly more complex than cars. Especially those with jet engines.

Airplanes of the same type flew (and still fly today!) in the same unit. WW2 German or modern American makes planes no difference. Operationally you can 'mix and match', for example a flight of Stuka's accompanied by BF 109. They can of course land and be maintained on the same airfield. Airfields are much larger than most people imagine.

On that airfield the BF 109 and the Stuka would have their own workshops, with their own specialized maintenance crews and logistics. You can't even change the pilots. A pilot is trained for specific types of aircraft. A BF 109 pilot would be useless in a Stuka and the other way around. At best, they could fly the other type, but no more than that.

They used Bf 109s in an air defense role in the vicinity of the field.

The Me 262 was due to low speed, very vulnerable in landing configuration and at ground.

It didn't help that the engines were thirsty and so to conserve fuel, the pilots were instructed to shut down as soon as possible and let tractors and special motorcycles pull the aircraft into its revetment.

While marauding RAF and USAAF fighters could intrude at any time.

Possibly yes but no concrete proof

I found this list of units operating Me-262 during the war. What we have mentioned is Erprobungskommando 262, Kommando Nowotny, Kommando Schenk (Conversion unit for bomber pilots), KG 51 (Kampfgeschwader 51 Staff unit, as well as Gruppen I and II), Kommando Edelweis (Experimental unit usin KG 51 pilots), Jagdgeschwader 7, Jagdgruppe 10 , III./Erganzungsjagdgeschwader/JG 2 (Me-262 training at Lechfeld), KG(J) 6, IV./Erganzungsjagdgeschwader/KG 51, Kommando Stamp (Experimental Night Fighter unit defending Berlin), I./KG(J)54 (Converted JU 88 unit) , Einsatz Kommando Braunegg (Reconnaissance unit), Nahaufklarungsgruppe 6 (formed from Kommando Braunegg), 1/Versuchsverband Ob. d.l. (special experimental unit), Jagdverband 44, I./Erganzungsjagdgeschwader/KG(J) (Training unit), II./Erganzungsjagdgeschwader/KG(J) (Training unit).

Casual connoisseurs of Luftwaffe would notice (beside well known units like Kommando Nowotny, JG 7 and JV 44) some interesting squadrons. For example III./Erganzungsjagdgeschwader/JG 2 . JG 2 is better known for its FW-190 but they did operate Bf-109 practically till the end of the war. What is more interesting is did aforementioned unit organizationally belonged to JG 2. Some sources actually mention it as a part of completely different Jagdgeschwader, i.e. Ergänzungs-Jagdgeschwader 2 . It is unclear was EJG 2 somehow related to JG 2. Anyway, it is unlikely that Me-262 of this unit operated together with Bf-109 except possibly on ad hoc basis.

Other possible candidate is Jagdgruppe 10 . This unit certainly operated FW-190. I could not find mention of Bf-109 but considering chaotic nature of the last months of the war, it could not be completely excluded.

KG 51 did have Ju-88, Me-410 and possibly Fw-190 (but this is disputed) but no mention of Bf-109 so we will exclude this unit. Same could be said about KG(J)54.

Nahaufklarungsgruppe 6 is another possibility but again with no concrete proof. It was reconnaissance unit, and such units were often equipped with specialized versions of Bf-109.

Finally, there is one rather weak possibility. It is well known that JV 44 had some FW-190 on hand as Platzshutzstaffel (Airfield Protection Squadron) in April 1945. However, it is quite possible that Bf-109 did protect its airfield before that. They were administratively from different units, but in these last months of the war odd ball unit like JV 44 was often subordinated to other, larger commands. Therefore, technically Bf-109 protecting the airfield and Me-262 form JV 44 could belong to same Luftwaffe unit operating in the area.