Whether other party could file cross objections against the appeal preferred by the assessee or by the department

On receipt of notice that an appeal the other party could prefer an cross objection to the appeal even though he has not preferred an appeal. A Memorandum of Cross Objections have to be filed within 45 days from the date of receipt of the notice of appeal in FORM GST APL-06. The Tribunal shall dispose of the cross objections as if it is an appeal

(ICAI FAQ PUBLICATIONS 06-09-2017 Appeals to the Appellate Tribunal: FAQ NO. 44)

Whether the Department can also file an appeal before the Tribunal against the orders of Appellate Authority or review authority

Yes. The department could also prefer an appeal before the Hon’ble Tribunal in FORM GST APL-07. The procedure is detailed below:
 The Commissioner may, on his own motion, or upon request from the
Commissioner of State tax or Commissioner of Union territory tax, call for and
examine the record of any order passed by the Appellate Authority or the
Revisional Authority under this Act or the State Goods and Services Tax Act or
the Union Territory Goods and Services Tax Act for the purpose of satisfying
himself as to the legality or propriety of the said order and
 He may, by order, direct any officer subordinate to him to apply to the Appellate Tribunal within six months from the date on which the said order has been passed for determination of such points arising out of the said order as may be specified by the Commissioner in his order.

(ICAI FAQ PUBLICATIONS 06-09-2017 Appeals to the Appellate Tribunal: FAQ NO. 43)

Whether the proceedings before the Appellate Tribunal shall be deemed to be judicial

Yes. All proceedings before the Appellate Tribunal shall be deemed to be judicial proceedings within the meaning of sections 193 and 228, and for the purposes of section 196 of the Indian Penal Code, and the Appellate Tribunal shall be deemed to be civil court for the purposes of section 195 and Chapter XXVI of the Code of Criminal Procedure, 1973.

(ICAI FAQ PUBLICATIONS 06 09-2017   Procedure before Appellate Tribunal: FAQ NO. 40)

Whether the orders of the Appellate Tribunal are enforceable and if yes how

Any order made by the Appellate Tribunal may be enforced by it in the same manner as if it were a decree made by a court in a suit pending therein, and it shall be lawful for the Appellate Tribunal to send for execution of its orders to the court within the local limits of whose jurisdiction:
(a) in the case of an order against a company, the registered office of the company is situated; or
(b) in the case of an order against any other person, the person concerned
voluntarily resides or carries on business or personally works for gain

(ICAI FAQ PUBLICATIONS 06 09-2017   Procedure before Appellate Tribunal: FAQ NO. 39)

Whether the Appellate Tribunal has the same powers as that of Civil Court

Yes, to some extent the Tribunal is conferred with the powers of a Civil Court. The Appellate Tribunal shall, for the purposes of discharging its functions under this Act, have the same powers as are vested in a civil court under the Code of Civil Procedure, 1908 while trying a suit in respect of the following matters, namely: –
(a) summoning and enforcing the attendance of any person and examining him on oath;
(b) requiring the discovery and production of documents;
(c) receiving evidence on affidavits;
(d) subject to the provisions of sections 123 and 124 of the Indian Evidence Act, 1872, requisitioning any public record or document or a copy of such record or document from any office;
(e) issuing commissions for the examination of witnesses or documents;
(f) dismissing a representation for default or deciding it ex parte;
(g) setting aside any order of dismissal of any representation for default or any order passed by it ex parte; and
(h) any other matter which may be prescribed.

(ICAI FAQ PUBLICATIONS 06 09-2017   Procedure before Appellate Tribunal: FAQ NO. 38)

Whether the Appellate Tribunal is bound by the procedures of Code of Civil Procedure

No. The Appellate Tribunal shall not, while disposing of any proceedings before it or an appeal before it, be bound by the procedure laid down in the Code of Civil Procedure, 1908, but shall be guided by the principles of natural justice and subject to the other provisions of this Act and the rules made there under, the Appellate Tribunal shall have power to regulate its own procedure.

(ICAI FAQ PUBLICATIONS 06-09-2017   Procedure before Appellate Tribunal: FAQ NO. 37)

Whether a President or member could be removed from the office

Yes.
A. National President, Members of National or Regional Tribunal The Central Government may, after consultation with the Chief Justice of India, in case of the President, Judicial Members and Technical Members of the National Bench, Regional Benches or Technical Members (Centre) of the State Bench or Area Benches, may remove from the office such President or Member, who—
(a) has been adjudged an insolvent; or
(b) has been convicted of an offence which, in the opinion of such Government
involves moral turpitude; or
(c) has become physically or mentally incapable of acting as such President, State President or Member; or
(d) has acquired such financial or other interest as is likely to affect prejudicially his functions as such President, State President or Member; or
(e) has so abused his position as to render his continuance in office prejudicial to the public interest:
B. State president and members of the State or area Tribunal:
Similarly, the State Government may, after consultation with the Chief Justice of High Court, in case of the State President, Judicial Members, Technical Members (State) of the State Bench or Area Benches may remove the member under the circumstances discussed at (a) to (e) above.
However, the President, State President or the Member shall not be removed on any of the grounds specified in clauses (d) and (e), unless he has been informed of the charges against him and has been given an opportunity of being heard.

(ICAI FAQ PUBLICATIONS 06-09-2017 Qualification, appointment & condition of service of the President and the members of the Tribunal: FAQ NO. 36)

For what period President and Members could hold the office

The President of the Appellate Tribunal shall hold office for a term of three years from the date on which he enters upon his office, or until he attains the age of seventy years, whichever is earlier and shall be eligible for reappointment The Judicial Member of the Appellate Tribunal and the State President shall hold office for a term of three years from the date on which he enters upon his office, or until he attains the age of sixty-five years, whichever is earlier and shall be eligible for reappointment.
The Technical Member (Centre) or Technical Member (State) of the Appellate Tribunal shall hold office for a term of five years from the date on which he enters upon his office, or until he attains the age of sixty-five years, whichever is earlier and shall be eligible for reappointment.

(ICAI FAQ PUBLICATIONS 06-09-2017 Qualification, appointment & condition of service of the President and the members of the Tribunal: FAQ NO. 35)