SQL Server: How to find position of the first uppercase non-ascii character in the string?

Question 1

If it really is just "firstname and lastname columns concatenation without a space between" then this may suffice:

select
    *
from mytable
where LEN(ConcatenatedName) <> LEN(REPLACE(ConcatenatedName, ' ', ''));

You can substitute whatever passes for a space in your local environment, of course. If there are, say, middle name which should be space-delimited this will pick them up too. However, you may get false positives if there are intended spaces e.g. two part surnames.

Second go

Fair point. I hadn't accounted for case. Here's some magic code which works with your test data:

with digits as
(
    SELECT
        *
    FROM
        (
        VALUES (0),(1), (2), (3), (4), (5), (6), (7), (8), (9)
        ) AS MyTable(i)
)
, Number as
(
    select (a.i * 10) + b.i as number
    from digits as a
    cross join digits as b
)
, LetterCase as
(
    select
        n.number
        ,t.value
        ,SUBSTRING(t.value, n.number, 1) as Letter
        ,ASCII(SUBSTRING(t.value, n.number, 1)) LetterASCII
        ,CASE
            when ASCII(SUBSTRING(t.value, n.number, 1)) between 65 and 90
                then 'True'
            else 'False'
        end as IsUpper
    from Number as n
    cross join #tmp as t
    where n.number between 1 and LEN(t.value)
)
select
    lc.value
from LetterCase as lc
where lc.IsUpper = 'True'
and lc.number > 1
and SUBSTRING(lc.value, lc.number - 1, 1) <> ' '

drop table #tmp;

It draws on my answer to this other question - Split words with a capital letter in sql.

Third go

Heres some magic code which works with your revised test data. You have to take care if your instance's (or database's or columns's) default collation is not the same as the one with which you want to work.

;with digits as
(
    SELECT
        *
    FROM
        (
        VALUES (0),(1), (2), (3), (4), (5), (6), (7), (8), (9)
        ) AS MyTable(i)
)
, Number as
(
    select (a.i * 10) + b.i as number
    from digits as a
    cross join digits as b
)
, UpperCaseCharacters as
(
    select NCHAR(1040) collate Cyrillic_General_CS_AI as CodePoint --А
    UNION ALL
    select NCHAR(1052) --М
    -- Extend this list with all the upper case character in your chosen glyph list.
)
, LetterCase as
(
    select
        n.number
        ,t.value
        ,CASE
            when SUBSTRING(t.value, n.number, 1) IN (select Codepoint from UpperCaseCharacters)
                then 'True'
            else ''
        end as IsUpper
    from Number as n
    cross join #tmp as t
    where n.number between 1 and LEN(t.value)
)
select
    lc.value
from LetterCase as lc
where lc.IsUpper = 'True'
and lc.number > 1
and SUBSTRING(lc.value, lc.number - 1, 1) <> ' ';

Question 2

You can use the basic pattern recognition of the LIKE operator.
The most used capability is the %, but it recognize set search with [set].
The pattern you search is

WHERE column LIKE '%[^ ][A-Z]%'

Where the character ^ is used to negate the following pattern.

EDIT
Checking the comment from the OP I found out that SQLFiddle use a case insensite collation, with that the check will never work, still I had to change the logic to "a lower case letter followed by something that is not a lower case letter or a space", but to do this the second range have to be expanded to add the space. For english letter is

WHERE column LIKE '%[a-z][^ abcdefghijklmnopqrstuvwxyz]%'

SQLFiddle demo

Question 3

select * from <Table_name> where <Field_name> COLLATE Latin1_General_CS_AS like '%bA%'

Question 4

The final solution in my case is TVF

alter function FindUpperCase(@value nvarchar(100) )
returns table
as return 
(
    with tmp as
    (
        select * from (values (@value)) as t(value)
    )
    , digits as
    (
        SELECT * FROM
            (
            VALUES (0),(1), (2), (3), (4), (5), (6), (7), (8), (9)
            ) AS MyTable(i)
    )
    , Number as
    (
        select (a.i * 10) + b.i as number
        from digits as a
        cross join digits as b
    )
    , LetterCase as
    (
        select
            n.number
            ,t.value
            ,CASE
                when UNICODE(SUBSTRING(t.value, n.number, 1)) BETWEEN UNICODE(N'А') AND UNICODE(N'Я')
                    then 'True'
                else ''
            end as IsUpper
        from Number as n
        cross join tmp as t
        where n.number between 1 and LEN(t.value)
    )
    select
        lc.value
    from LetterCase as lc
    where lc.IsUpper = 'True'
    and lc.number > 1
    and UNICODE(SUBSTRING(lc.value, lc.number - 1, 1)) BETWEEN UNICODE(N'а') AND UNICODE(N'я')
)

Usage

select c.fullname
from dbo.contacts c
cross apply dbo.FindUpperCase(c.fullname) as f