Importance of aggregating distributed energy resources (DERs) capacity to procure high-performance frequency regulation services ensuring adequate flexibility for renewable-rich power systems is increasing. Meanwhile, disregarding response performance variety arising from different technical characteristics and inherent uncertainties of DERs managed by an aggregator adversely impacts aggregated response (AR) from three viewpoints: technical feasibility of AR delivery, economic profit-making and fairness of capacity allocation. To tackle these challenges, this paper introduces a new probabilistic flexibility response index (FRI) to integrally evaluate DERs' performance considering four factors: structure, control, communication, aggregation and uncertainties and interactions of these factors. Also, a multi-objective stochastic bidding strategy framework in the day-ahead energy and regulation markets is developed, in which the DERs capacity allocation is devised using the evaluation of their FRIs as the second objective. This framework lies within a mixed-integer linear programming model, considering uncertainties of resources, communication systems, AR, and market prices. The multi-objective structure of problem is handled through augmented $\varepsilon $ -constraint and Fuzzy decision-making approaches. To validate the efficiency, scalability, and generality of the developed framework, it is implemented on three aggregators of different numbers and types of DERs participating in the ERCOT market context. The results show fundamental improvements regarding three viewpoints.