Hello Timber, welcome here to the forum.

Actually you should not be following the procedures for spouse/partner visa.

The EEA route is completely different in every respect.

In principle there is no need for the EEA national to already be in the UK, to have an NI number, to have money, to have suitable accommodation, to have been married for more than a day or to have the spouse pass any English Language test, or in fact to make any application fee for an EEA family permit.

In assessing an application, the ECO should be satisfied that:

1.the applicant is the family member of the EEA national (marriage certificate, birth certificate or other evidence of family link)
2.the EEA national is residing in the UK in accordance with the EEA Regulations (as qualified person if more than 3 months) and the non-EEA national is
joining them
OR
the EEA national intends to travel to the UK within 6 months and will have a right to reside under the Regulations on arrival, and the non-EEA national will be accompanying or joining the EEA national; and
3.if applying as a spouse or civil partner, there are no grounds to consider that the marriage or civil partnership is one of convenience (see Annex ….); and
4.if applying as dependent family members (dependent children 21 and over and dependent relatives) they are dependent on the EEA national or the EEA national’s spouse or civil partner; and
5.neither the applicant nor the EEA national should be excluded from the UK on the grounds of public policy, public security or public health.

So unless you have any special reasons why the above does not apply to you then it should be an exceptionally easy and stress free process.

Wish it was the same for British Citizens living in UK

As a first step please review this UKBA page specifically for Family Permits - How to apply for an EEA family permit
Please do follow all the links provided.
May I also suggest you review the VAF5 application form and VAF2-8B Guide, both of which can be downloaded from the Right-Hand side on the UKBA webpage.

After your review if you have any specific question about your specific circumstances just fire away here.

So my questions and concerns are the following:
- First of all...
= Do we need to 'inform' any office here in the UK about our marriage? I have an original copy of the Marriage Certificate.
No need as that will be covered under the FP application.

- As I am being a EEA national we can apply for the EEA FP.
Correct.

= Should I apply in the UK for her Entry Clearance or should she apply for EEA FP in Philippines?
She is the one to apply for the EEA FP. This needs to be in the Philippines and via the VFS online system

= What are the costs involved in one or the other way and what are the usual average processing times?
The Family Permit is completely free of any charges. Please see the UKBA webpage to access processing times. Usually it's just a few weeks as FP's are given priority and also there are very few checks to be made by the ECO's and very few reasons allowed for refusal.

= With my shared accomodation, my not too high (average Net 1000 PPM) income do we have a chance?
There is no income requirement to be met when applying for FP

= With the quite big age gap (18 years) and she being quite young (19 years old) could we face problems?
No problems


= I am receiving Working Tax Credit. Is it considered as a Benefit or kind of dependancy on state funds?
In so far as you are legally allowed such benefits there is no problem.

- Once we have the EEA FP...
= Should I travel to the Philippines to 'pick her up'? I am just very worried if anything happens at the UK border she would be alone... but financially it's not easy...
Of course that is your personal choice. I would expect you to be present at the UK airport to pick her up when she arrives, so UKBA would be able to contact you in the highly unlikely event that they need to.

Can she apply for an NI straight away?
Yes she can, but I strongly suggests the first thing she applies for is her Residence Permit/Card.
It's often difficult to prove eligibility to work without that. Once she has made application she will receive confirmation of that from UKBA by letter.
The actual Residence Permit may take up to 6 months.

= How long she can stay without the need of renew the 'Visa'?
The Family Permit is NOT a visa, it is a form of entry clearance. Normally it has a validity of 6 months, which means she should enter UK with that validity period.
Once she is here in UK she does not need to do anything. Although as I suggested before it is so much better she applies Residence.
She can stay here in UK under her Residence Card for 5 years. After that she may apply for Permanent residence then after 12 months for British Citizenship.

= Can she travel within the EEA freely if she travels with me?
Technically yes, but she needs to have acceptable travel documents and ID.
Without a schenghen visa and/or residence card she will face problems at airports and may be refused boarding. She may face problems both within EU borders and also
re-entering UK. FP is a one time entry clearance.
Better to ask that question again once she is here

= Does she needs any extra permit to start working?
Technically no. But in practice plenty of people had lots of trouble convincing employers of their status. Employers need to see documentary proof of
eligibility to work here. That's why she should apply Residence Card asap.

Hope this helps