Where an assessee gets a favourable decision either before Appellate Authority or the Tribunal, whether the pre-deposit would be refunded back? Whether any interest is paid

Yes, where the assessee gets a favourable decision (fully or partially), the amount paid as pre-deposit would be refunded along with interest which is computed from the date of deposit till date of refund.
Further, Section 115 provides that interest at the rate specified under section 56 shall be payable in respect of such refund from the date of payment of the amount till the date of refund of such amount.

(ICAI FAQ PUBLICATIONS 06-09-2017 ORDERS OF APPELLATE TRIBUNAL: FAQ NO. 54)

Whether Tribunal has power to rectify / amend the orders passed by it

Yes, Tribunal may amend any order passed by it under in terms of Section 113(1) so as to rectify any mistake apparent from the record. Tribunal could undertake rectification on its own or on application by either of the parties to the appeal. The application for rectification shall be made within a period of three months from the date of the order sought to be rectified.
However, no amendment which has the effect of enhancing an assessment or reducing a refund or input tax credit or otherwise increasing the liability of the other party, shall be made under this sub-section, unless the Tribunal has given notice to him of its intention to do so and has allowed him a reasonable opportunity of being heard.

(ICAI FAQ PUBLICATIONS 06-09-2017 ORDERS OF APPELLATE TRIBUNAL: FAQ NO. 53)

What are the orders that could be passed by the Appellate Tribunal

The Hon’ble Tribunal may pass such orders thereon as it thinks fit, confirming, modifying or annulling the decision or order appealed against or may refer the case back to the Appellate Authority, or the Revisional Authority or to the original adjudicating authority, with such directions as it may think fit, for a fresh adjudication or decision after taking additional evidence, if necessary.

(ICAI FAQ PUBLICATIONS 06-09-2017 ORDERS OF APPELLATE TRIBUNAL: FAQ NO. 52)

Whether the parties to an appeal could seek adjournment of the hearing

The Tribunal may, if sufficient cause is shown at any stage of hearing of an appeal, grant time, from time to time, to the parties or any of them and adjourn the hearing of the appeal for reasons to be recorded in writing. However, no such adjournment shall be granted more than three times to a party during hearing of the appeal.

(ICAI FAQ PUBLICATIONS 06-09-2017 Orders of Appellate Tribunal: FAQ NO. 51)

Whether entire amount of tax, interest, penalty, fine or fee has to be remitted before preferring appeal

For filing appeal following amounts shall have to be remitted as pre-deposit.
(a) Where any part of the amount of tax, interest, fine, fee and penalty arising from the impugned order, as is admitted by him, entire such amount and
(b) a sum equal to 25% of the remaining amount of tax in dispute arising from the said order, in relation to which the appeal has been filed shall be paid. This
amount (25%) is in addition to the pre-deposit paid while filing appeal before
Appellate Authority.

(ICAI FAQ PUBLICATIONS 06-09-2017 Appeals to the Appellate Tribunal: FAQ NO. 47)

Whether appeal / application / cross objections filed beyond the time limit would be entertained

Tribunal has been conferred with powers to condone the delay upto 3 months, beyond the period of 3 months or 6 months in case of filing of appeals, where sufficient cause for the delay is shown.
Similarly, delay upto 45 days could be condoned by the Tribunal in filing the
memorandum of cross objections where sufficient cause for the delay is shown.

(ICAI FAQ PUBLICATIONS 06-09-2017 Appeals to the Appellate Tribunal: FAQ NO. 46)

What is the time limit for filing appeal / cross objections

Appeal by Assessee: 3 months from the date of receipt of the order against which the appeal is being preferred
Application by Department: 6 months from the date of receipt of the order against which the application is being preferred
Memorandum of Cross-Objections: 45 days from the date of receipt of notice of appeal

(ICAI FAQ PUBLICATIONS 06-09-2017 Appeals to the Appellate Tribunal: FAQ NO. 45)