Depends on the nature of the exercises. Some will want you to travel to the UK, others may be able to adapt the assessments so they can be done remotely.
You won't need a visa for an interview. They may pay your air fare, but typically that is a capped amount. Even for the more generous of firms, I would expect that to be capped at £300, so less likely to cover all of your expenses.
When applying to a firm, the first thing to check with them is whether they can provide you with a visa for the summer internship. This won't be straight forward for many firms and can be very costly compared to the cost of employing you for 2-4 weeks.