Will a Verizon iPhone 5 be able to utilize T-Mobile's 4G and LTE?

Yes if you are located in a t-mobile refarmed area for "4G". No LTE for t-mobile yet.

 

If your iPhone 5 is unlocked and on registered on T-Mobile then yes. but LTE has not be brought. Only 4G.

P.S. whatever you do, by no circumstance, do not go to AT&T.

 

Both AT&T unlocked i5 and verizon i5 will get HPSA+ that Tmobile calls it 4G.
Only the AT&T or factory unlocked iphone will be able to access Tmobiles LTE if it ever goes online. The verizon i5 will only get LTE for verizon and outside the US with international carriers that support those bands.

Yes, DO NOT go to AT&T or You might get LTE speeds like this below
Go to Tmobile and stick with 2G edge data.