Stars at near break-up rotation rates
I have not the qualification to answer the question in its whole but the question is interesting (I worked on Be Stars which are episodically surrounded by an decretion disk and which rotates at nearly critical velocities. The phenomenon in Be stars is different from accreting stars. The only consequences of subcritical velocity is a flattened envelope and modification of its inner structure and of the oscillations modes found in these stars
(if you have time and curiosity, a good example of flattened star with Keplerian rotating decretion disk is Achernar, a Be star observed using interferometry --> Have a look at Meilland et al. 2007: www.aanda.org/articles/aa/pdf/2007/10/aa4848-06.pdf)

Anyway...

I found this paper about critically-rotating accretors. May be you will find answers to your questions here or in its references (use nasa ads site for your query: http://adsabs.harvard.edu/).
http: //arxiv.org/pdf/1306.1348v2.pdf
It seems that in the introduction, there is some answers to your questions about reaching the critical velocity.

The accreted mass can increase the rotation rate until the star reaches the critical velocity.

It is said : "For a typical 6 + 3.6 M⊙ system, with initial period
Pinit = 2.5 days, in the absence of spin-down mechanisms, only
3 per cent (0.12 M⊙) of the total amount of matter transferred
by RLOF (more than 5 M⊙) is enough to spin the gainer up
to the critical rotation."

But we still don't know if the gainer can really reach the critical velocity. Some papers are dealing with break-up mechanims which does not allow the gainer to reach critical velocity: spin down by tides, magnetic breaking, limitation of the accretion angular momentum through interaction with the accretion disk, stopping of the accretion mechanism...

I am sure that you will find many papers on nasa ads that will give you answers to your questions.