Upgrading garage fuse box to breaker box using existing 10/2 wire with ground
Given your fairly precisely stated 15/16" OD of the pipe, the only pipe anywhere near that size is EMT "Electrical Metal Tubing".  But you say it's plastic.  I can't find any plastic pipe near that dimension. Everything is &lt;7/8" or well over 1", nearly 1-1/16".  I'll proceed assuming EMT, since my wire reco has a lot of slack. 

It also sounds like you consider the 30A an undesired restriction, and would prefer to exceed it.  That's just as well, because the cables now in the pipe are totally unworkable, and must be removed.  I won't get into a codevio list, because that would be tiring, insult your father's work, and just cheese you off. Suffice it to say, there is no way I can possibly see to make use of the NM cables in the pipe. 

You can run up to #6 copper, for 70 amps.

That 3/4" pipe will support up to 6 AWG wire - that is to say, three #6 THWN-2 individual wires, and one #8 bare ground wire. (even though EMT is a valid grounding path, I would not trust it outdoors).  That will put you at 29% fill, which is perfectly reasonable for 4 wires in a pipe.

(this works for all pipes except 3/4" schedule 80 PVC; for that, a #8 ground wire won't fit due to conduit fill limitations; the #10 ground limits you to 60A. But then, if it was schedule 80, I supremely doubt you got two #10 cables in there!) 

That means you'll be able to support the load you have planned, plus a pretty good sized EVSE, wood shop, whatever will come up for you. 

Given the short run of the wire, I don't see any reason to bother with aluminum wire (#6 Al would limit you to 50A) or with shrinking the wire to save a few bucks.  Having 70A service at the garage would be sweet. 

The 10/2 NM cables have to go

Unfortunately they will be annoyingly difficult to pull. This is what you are up against.  

2 x 10-2 inside EMT: 
2 x 10-2 inside Sched 80 PVC: 

It's actually worse than it looks in the drawing. It'll be bind-up city. This is why oval cables are treated as a round wire of the large dimension. 

But steady on; get that wire outta there, and the three #6 will go in happy as a lark.  Or smaller wire will obviously be easier still. 



As to your questions:

Feed breaker size

What breaker for the existing 10-2 cable?  None.  That cable can't be supported for a variety of reasons. 

What breaker for new 3x#10 THHN wires?  30A.  Only. Because of NEC 240.4(D). 

What breaker for larger THHN wires?  Since you are 100% in conduit, and using THWN-2 wires, look at the 75 degree C column in Table 310.15(B)(16) and round up to the next larger breaker.  So 8Cu/6Al = 50A.  6Cu = 65->70A.   Larger wires won't fit. 

Subpanel size: No place to scrimp!

What size subpanel is appropriate for a 10 AWG feed?  Subpanels are about breaker spaces, first and foremost. That is the defining characteristic of a panel.  A lot of people decide they want to scrimp on the panel spaces, so they get a smaller panel.  This saves them the price ... of a latté.  Then later, they have a $1000 problem because the panel is full.  So gospel around here is Think Big. Think at least 16-18 space, and a lot more if you reasonably expect to fill that.  Spaces are cheap, in fact, they make other things cheaper.  Look for combo packs which include a main breaker and some "bonus breakers" to save yourself some money.

The second characteristic is bus amperage / main breaker trip. "Going Big" probably mooted that, as we're way beyond the 30A-bussed or 60A-bussed micro-panels.  So if your panel has a 100A main breaker, or a 125A bus, that'll work out just fine.   The main breaker is only there to be a disconnect switch (since this is an outbuilding), it's not there to be a breaker, so we don't care what size it is.

Hurry up and pull that permit before NEC 2020 kicks; that'll require an exterior disconnect switch also. If you're in Massachusetts, too late.   

"But I want the nearby breaker to be the same size so it'll trip locally instead of at the house!" That trick never works,  and you shouldn't even be trying.  First, both breakers see exactly the same amount of current, so it would be a 50/50 chance even if they were identical. Second, they're not identical - main breakers tend to have a more forgiving trip curve.  Third, if that happens often enough to care, the feeder is too small and must be enlarged. (see why I'm recommending #6? :) Feeder must be at 25% above expected practical load.