A1: Without Tessaiga, Inuyasha may not be able to hold on much longer, unless his demon form is awakened. If the latter happens, he would
probably definitely win over Koga.
A2: So long as Inuyasha made an effort to take Kouga's shards from him, he could win. Evidence being Inuyasha's fight with Bankotsu, who was waaaay tougher than Kouga. Also, when Naraku tricked Kouga into fighting Inuyasha, Inuyasha was able to hold his own for quite a while before he tried using Tessaiga. Plus he was holding back because he knew Kouga was being used.