بسم الله الرحمن الرحيم
(1) The itikaf of last ten days of Ramadhan is Sunnat Muakkadah Kifaya. In that itikaf, one cannot go outside for bath except the obligatory ghusl. Yes, if one observes the nafl itikaf of first or second ten days, then he or she can take bath of Jumuah or just to get fresh.
(2) What do you mean by “offering with mother”? If you mean that you offer salah in the same room with your mother where you are staying in itikaf, then there is no harm, rather it is lawful without any desirability (makrooh). But if you mean that you lead (do imamat) of the tarawih salah and your mother prays behind you, then this is makrooh even if it is tarawih. The women should offer tarawih individually like the daily five time salah. The same ruling applies you even though you are hafiza.
(3) If you and your mother are in itikaf, then both can stay in the same room. Itikaf will be nullified if you switch from that room to other. If both of you are doing itikaf in two different rooms, then both of you should stay in your own rooms, it is not lawful to move from your room to other’s room, it will nullify itikaf.
(4) There is no harm if the maid etc comes to your room; rather your mahram men like father, real chacha, real brothers and real nephews also can see you. But if you fear that they will involve you in nonsense talks, you can also stop them.
Allah knows Best!
Darul Uloom Deoband