Effect of short selling the stock on the buying power and cash position

If you have $25k cash in your account and the margin offered your broker is 50% then you have $50k of buying power (not $75k). At 50%, your broker will lend you an amount equal to the total of your cash and marginable securities. Here's the long margin calculation:

 Market     Debit    Equity   Margin 
 Value
        
 50,000     25,000   25,000   50.00%

I short sale stock worth $5000. How much of initial cash is now available for buying securities after shorting?

The Reg T short margin requirement is 50% of the value of the short sale ($2.5k). That leaves $47.5k for margin buying.

The shorted stock is now worth $10,000. How much of initial cash is now available for buying securities? What would be my buying power now? Would would be my margin amount available now?

Reg T margin maintenance requirement for a short position is 30%. At $10k, that's $3k maintenance. Your short position lost $5k so account value dropped to $20k ($40k BP) and with the MMR of $3k, you'd have $37k available for long margin buying.

Another way of looking at this is that you started with $50k of buying power. The initial margin for your short is $2.5k. When the short doubles to $10k, you have to add another $5.5k to meet MMR. That's $8k. With the $5k loss of equity, that's $13k. $50k - $13k equals $37k of buying power.

 Short Sale    Initial    Total 
    Value      Margin     Marg Req  
        
   5,000       2,500     7,500 

        
 Short Sale    30% Maint   Total     Margin
   Value       Margin     Marg Req    Call
        
  10,000       3,000     13,000      5,500
        

Note that brokers can require more margin than Reg T.