Numbers made from digits 1-9 -- proving the exceptions?

Question

Inspired by this paper

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Introduction

In this paper, a sequential representation of a number is a formula that uses the digits 1-9 in order with the mathematical operations +, -, ×, ÷, ^, as well as brackets, and digit-to-digit concatenation.

For example, 100 can be represented as follows: $$100 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 × 9$$ More complicated formulae look more like this: $$11106 = (1 + 2^3)^4 + 567 × 8 + 9$$ $$1+i = (-1)^{(2+3+4)/(5+6+7)}-8+9$$

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Question

Curiously enough, a solution the number 10958 has not been found*, despite extensive searching and computing. It was even touched upon in a numberphile video.

Can we prove certain numbers must or must not have a sequential representation?

*Cited as "still not available" in the paper

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Related questions

Note: don't strain yourself, they're not as important.

1. What is the trivial minimum/maximum bounds with sequential representation?

2. Can we answer these questions with more simple rules? Such as removing concatenation or exponentiation. Or perhaps the rule of digit order?

3. Since there is a finite amount of combinations one can make with these digits and operations, how many combinations are there?

4. How many unique numbers have a sequential representation?