Yes, a determination of responsibility is required for all those awards. Your answer is at FAR 9.103 (b) 9.103 - Policy.
(a) Purchases shall be made from, and contracts shall be awarded to, responsible prospective contractors only.
(b) (b) No purchase or award shall be made unless the contracting officer makes an affirmative determination of responsibility. In the absence of information clearly indicating that the prospective contractor is responsible, the contracting officer shall make a determination of non responsibility. If the prospective contractor is a small business concern, the contracting officer shall comply with Subpart 19.6, should not make any award or purchase without to a responsibility determination.
There is no exception for furniture or other commodities or small businesses or task/delivery orders. FAR 19.6 lays out some special rules if the contracting officer believes that a small business (including an 8(a)) isn't responsible. But a non-responsibility determination has to be made before you go to SBA for a Certificate of Competency.
If you just think about it, this rule makes sense. We clearly want to check all contractors to see if they haven't been put on a debarred or suspended list, for example, before we make award. On-line systems such as FAPIIS are making this easier to do.